2594d8905c
Rewrote some of the chapter examples to use animated plots. Makes it so much easier to see how the filter is performing.
602 lines
19 KiB
Python
602 lines
19 KiB
Python
# -*- coding: utf-8 -*-
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"""Copyright 2015 Roger R Labbe Jr.
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Code supporting the book
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Kalman and Bayesian Filters in Python
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https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python
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This is licensed under an MIT license. See the LICENSE.txt file
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for more information.
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"""
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from __future__ import (absolute_import, division, print_function,
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unicode_literals)
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import matplotlib.pyplot as plt
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import numpy as np
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from contextlib import contextmanager
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import sys
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sys.path.insert(0, '..')
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from book_format import figsize
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try:
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import seaborn
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except:
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pass
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def end_interactive(fig):
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""" end interaction in a plot created with %matplotlib notebook """
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import time
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plt.gcf().canvas.draw()
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time.sleep(0.1)
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plt.close(fig)
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@contextmanager
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def interactive_plot(close=True, fig=None):
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if fig is None:
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fig = plt.figure()
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yield
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plt.tight_layout()
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if close:
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end_interactive(fig)
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def plot_errorbars(bars, xlims, ylims=(0, 2)):
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i = 0.0
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for bar in bars:
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plt.errorbar([bar[0]], [i], xerr=[bar[1]], fmt='o', label=bar[2] , capthick=2, capsize=10)
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i += 0.2
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plt.ylim(*ylims)
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plt.xlim(xlims[0], xlims[1])
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show_legend()
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plt.gca().axes.yaxis.set_ticks([])
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plt.show()
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def plot_errorbar1():
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with figsize(y=2):
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plt.figure()
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plot_errorbars([(160, 8, 'A'), (170, 8, 'B')],
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xlims=(145, 185), ylims=(-1, 1))
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plt.show()
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plt.savefig('../figs/gh_errorbar1.png', pad_inches=0.)
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def plot_errorbar2():
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with figsize(y=2):
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plt.figure()
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plot_errorbars([(160, 3, 'A'), (170, 9, 'B')],
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xlims=(145, 185), ylims=(-1, 1))
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plt.savefig('../figs/gh_errorbar2.png', pad_inches=0.)
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def plot_errorbar3():
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with figsize(y=2):
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plt.figure()
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plot_errorbars([(160, 1, 'A'), (170, 9, 'B')],
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xlims=(145, 185), ylims=(-1, 1))
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plt.savefig('../figs/gh_errorbar3.png', pad_inches=0.1)
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def plot_hypothesis1():
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with figsize(y=2.5):
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plt.figure()
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plt.errorbar([1, 2, 3], [170, 161, 169],
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xerr=0, yerr=10, fmt='bo', capthick=2, capsize=10)
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plt.plot([1, 3], [180, 160], color='g', ls='--')
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plt.plot([1, 3], [170, 170], color='g', ls='--')
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plt.plot([1, 3], [160, 175], color='g', ls='--')
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plt.plot([1, 2, 3], [180, 152, 179], color='g', ls='--')
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plt.xlim(0,4); plt.ylim(150, 185)
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plt.xlabel('day')
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plt.ylabel('lbs')
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plt.tight_layout()
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plt.savefig('../figs/gh_hypothesis1.png', pad_inches=0.1)
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def plot_hypothesis2():
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with figsize(y=2.5):
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plt.figure()
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plt.errorbar(range(1, 11), [169, 170, 169,171, 170, 171, 169, 170, 169, 170],
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xerr=0, yerr=6, fmt='bo', capthick=2, capsize=10)
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plt.plot([1, 10], [169, 170.5], color='g', ls='--')
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plt.xlim(0, 11); plt.ylim(150, 185)
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plt.xlabel('day')
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plt.ylabel('lbs')
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plt.savefig('../figs/gh_hypothesis2.png', pad_inches=0.1)
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def plot_hypothesis3():
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weights = [158.0, 164.2, 160.3, 159.9, 162.1, 164.6,
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169.6, 167.4, 166.4, 171.0, 171.2, 172.6]
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with figsize(y=2.5):
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plt.figure()
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plt.errorbar(range(1, 13), weights,
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xerr=0, yerr=6, fmt='o', capthick=2, capsize=10)
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plt.xlim(0, 13); plt.ylim(145, 185)
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plt.xlabel('day')
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plt.ylabel('weight (lbs)')
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plt.savefig('../figs/gh_hypothesis3.png', pad_inches=0.1)
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def plot_hypothesis4():
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weights = [158.0, 164.2, 160.3, 159.9, 162.1, 164.6,
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169.6, 167.4, 166.4, 171.0, 171.2, 172.6]
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with figsize(y=2.5):
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plt.figure()
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ave = np.sum(weights) / len(weights)
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plt.errorbar(range(1,13), weights, label='weights',
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yerr=6, fmt='o', capthick=2, capsize=10)
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plt.plot([1, 12], [ave,ave], c='r', label='hypothesis')
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plt.xlim(0, 13); plt.ylim(145, 185)
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plt.xlabel('day')
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plt.ylabel('weight (lbs)')
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show_legend()
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plt.savefig('../figs/gh_hypothesis4.png', pad_inches=0.1)
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def plot_hypothesis5():
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weights = [158.0, 164.2, 160.3, 159.9, 162.1, 164.6,
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169.6, 167.4, 166.4, 171.0, 171.2, 172.6]
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xs = range(1, len(weights)+1)
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line = np.poly1d(np.polyfit(xs, weights, 1))
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with figsize(y=2.5):
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plt.figure()
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plt.errorbar(range(1, 13), weights, label='weights',
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yerr=5, fmt='o', capthick=2, capsize=10)
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plt.plot (xs, line(xs), c='r', label='hypothesis')
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plt.xlim(0, 13); plt.ylim(145, 185)
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plt.xlabel('day')
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plt.ylabel('weight (lbs)')
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show_legend()
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plt.savefig('../figs/gh_hypothesis5.png', pad_inches=0.1)
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def plot_estimate_chart_1():
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with figsize(y=2.5):
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plt.figure()
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ax = plt.axes()
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ax.annotate('', xy=[1,159], xytext=[0,158],
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arrowprops=dict(arrowstyle='->', ec='r',shrinkA=6, lw=3,shrinkB=5))
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plt.scatter ([0], [158], c='b')
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plt.scatter ([1], [159], c='r')
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plt.xlabel('day')
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plt.ylabel('weight (lbs)')
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ax.xaxis.grid(True, which="major", linestyle='dotted')
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ax.yaxis.grid(True, which="major", linestyle='dotted')
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plt.tight_layout()
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plt.savefig('../figs/gh_estimate1.png', pad_inches=0.1)
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def plot_estimate_chart_2():
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with figsize(y=2.5):
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plt.figure()
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ax = plt.axes()
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ax.annotate('', xy=[1,159], xytext=[0,158],
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arrowprops=dict(arrowstyle='->',
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ec='r', lw=3, shrinkA=6, shrinkB=5))
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plt.scatter ([0], [158.0], c='k',s=128)
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plt.scatter ([1], [164.2], c='b',s=128)
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plt.scatter ([1], [159], c='r', s=128)
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plt.text (1.0, 158.8, "prediction ($x_t)$", ha='center',va='top',fontsize=18,color='red')
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plt.text (1.0, 164.4, "measurement ($z$)",ha='center',va='bottom',fontsize=18,color='blue')
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plt.text (0, 157.8, "estimate ($\hat{x}_{t-1}$)", ha='center', va='top',fontsize=18)
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plt.xlabel('day')
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plt.ylabel('weight (lbs)')
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ax.xaxis.grid(True, which="major", linestyle='dotted')
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ax.yaxis.grid(True, which="major", linestyle='dotted')
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plt.savefig('../figs/gh_estimate2.png', pad_inches=0.1)
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def plot_estimate_chart_3():
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with figsize(y=2.5):
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plt.figure()
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ax = plt.axes()
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ax.annotate('', xy=[1,159], xytext=[0,158],
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arrowprops=dict(arrowstyle='->',
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ec='r', lw=3, shrinkA=6, shrinkB=5))
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ax.annotate('', xy=[1,159], xytext=[1,164.2],
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arrowprops=dict(arrowstyle='-',
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ec='k', lw=3, shrinkA=8, shrinkB=8))
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est_y = ((164.2-158)*.8 + 158)
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plt.scatter ([0,1], [158.0,est_y], c='k',s=128)
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plt.scatter ([1], [164.2], c='b',s=128)
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plt.scatter ([1], [159], c='r', s=128)
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plt.text (1.0, 158.8, "prediction ($x_t)$", ha='center',va='top',fontsize=18,color='red')
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plt.text (1.0, 164.4, "measurement ($z$)",ha='center',va='bottom',fontsize=18,color='blue')
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plt.text (0, 157.8, "estimate ($\hat{x}_{t-1}$)", ha='center', va='top',fontsize=18)
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plt.text (0.95, est_y, "new estimate ($\hat{x}_{t}$)", ha='right', va='center',fontsize=18)
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plt.xlabel('day')
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plt.ylabel('weight (lbs)')
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ax.xaxis.grid(True, which="major", linestyle='dotted')
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ax.yaxis.grid(True, which="major", linestyle='dotted')
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plt.savefig('../figs/gh_estimate3.png', pad_inches=0.1)
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def create_predict_update_chart(box_bg = '#CCCCCC',
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arrow1 = '#88CCFF',
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arrow2 = '#88FF88'):
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plt.figure(figsize=(4, 2.), facecolor='w')
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#plt.figure(figsize=(14,12.5), facecolor='w')
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ax = plt.axes((0, 0, 1, 1),
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xticks=[], yticks=[], frameon=False)
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pc = Circle((4,5), 0.7, fc=box_bg)
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uc = Circle((6,5), 0.7, fc=box_bg)
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ax.add_patch (pc)
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ax.add_patch (uc)
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plt.text(4,5, "Predict\nStep",ha='center', va='center', fontsize=12)
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plt.text(6,5, "Update\nStep",ha='center', va='center', fontsize=12)
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#btm arrow from update to predict
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ax.annotate('',
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xy=(4.1, 4.5), xycoords='data',
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xytext=(6, 4.5), textcoords='data',
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size=20,
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arrowprops=dict(arrowstyle="simple",
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fc="0.6", ec="none",
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patchB=pc,
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patchA=uc,
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connectionstyle="arc3,rad=-0.5"))
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#top arrow from predict to update
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ax.annotate('',
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xy=(6, 5.5), xycoords='data',
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xytext=(4.1, 5.5), textcoords='data',
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size=20,
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arrowprops=dict(arrowstyle="simple",
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fc="0.6", ec="none",
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patchB=uc,
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patchA=pc,
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connectionstyle="arc3,rad=-0.5"))
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ax.annotate('Measurement ($\mathbf{z_k}$)',
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xy=(6.3, 5.6), xycoords='data',
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xytext=(6,6), textcoords='data',
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size=14,
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arrowprops=dict(arrowstyle="simple",
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fc="0.6", ec="none"))
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# arrow from predict to state estimate
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ax.annotate('',
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xy=(4.0, 3.8), xycoords='data',
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xytext=(4.0,4.3), textcoords='data',
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size=12,
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arrowprops=dict(arrowstyle="simple",
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fc="0.6", ec="none"))
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ax.annotate('Initial\nConditions ($\mathbf{x_0}$)',
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xy=(4.05, 5.7), xycoords='data',
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xytext=(2.5, 6.5), textcoords='data',
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size=14,
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arrowprops=dict(arrowstyle="simple",
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fc="0.6", ec="none"))
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plt.text (4, 3.7,'State Estimate ($\mathbf{\hat{x}_k}$)',
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ha='center', va='center', fontsize=14)
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plt.axis('equal')
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plt.xlim(2,10)
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plt.savefig('../figs/gh_predict_update.png', pad_inches=0.1)
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def show_residual_chart(show_eq=True, show_H=False):
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plt.figure(figsize=(11, 3.), facecolor='w')
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est_y = ((164.2-158)*.8 + 158)
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ax = plt.axes(xticks=[], yticks=[], frameon=False)
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ax.annotate('', xy=[1,159], xytext=[0,158],
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arrowprops=dict(arrowstyle='->',
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ec='r', lw=3, shrinkA=6, shrinkB=5))
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ax.annotate('', xy=[1,159], xytext=[1,164.2],
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arrowprops=dict(arrowstyle='-',
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ec='k', lw=3, shrinkA=8, shrinkB=8))
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ax.annotate('', xy=(1., est_y), xytext=(0.9, est_y),
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arrowprops=dict(arrowstyle='->', ec='#004080',
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lw=2,
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shrinkA=3, shrinkB=4))
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plt.scatter ([0,1], [158.0,est_y], c='k',s=128)
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plt.scatter ([1], [164.2], c='b',s=128)
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plt.scatter ([1], [159], c='r', s=128)
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plt.text (1.05, 158.8, r"prior $(\bar{x}_t)$", ha='center',va='top',fontsize=18,color='red')
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plt.text (0.5, 159.6, "prediction", ha='center',va='top',fontsize=18,color='red')
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plt.text (1.0, 164.4, r"measurement ($z$)",ha='center',va='bottom',fontsize=18,color='blue')
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plt.text (0, 157.8, r"posterior ($x_{t-1}$)", ha='center', va='top',fontsize=18)
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plt.text (1.02, est_y-1.5, "residual($y$)", ha='left', va='center',fontsize=18)
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if show_eq:
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if show_H:
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plt.text (1.02, est_y-2.2, r"$y=z-H\bar x_t$", ha='left', va='center',fontsize=18)
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else:
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plt.text (1.02, est_y-2.2, r"$y=z-\bar x_t$", ha='left', va='center',fontsize=18)
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plt.text (0.9, est_y, "new estimate ($x_t$)", ha='right', va='center',fontsize=18)
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plt.text (0.8, est_y-0.5, "(posterior)", ha='right', va='center',fontsize=18)
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if show_eq:
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plt.text (0.75, est_y-1.2, r"$\bar{x}_t + Ky$", ha='right', va='center',fontsize=18)
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plt.xlabel('time')
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ax.yaxis.set_label_position("right")
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plt.ylabel('state')
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plt.xlim(-0.1, 1.5)
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if show_H:
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plt.savefig('../figs/residual_chart_with_h.png', pad_inches=0.1)
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else:
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plt.savefig('../figs/residual_chart.png', pad_inches=0.1)
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def show_legend():
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plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
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def bar_plot(pos, x=None, ylim=(0,1), title=None, c='#30a2da',
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**kwargs):
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""" plot the values in `pos` as a bar plot.
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**Parameters**
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pos : list-like
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list of values to plot as bars
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x : list-like, optional
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If provided, specifies the x value for each value in pos. If not
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provided, the first pos element is plotted at x == 0, the second
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at 1, etc.
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ylim : (lower, upper), default = (0,1)
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specifies the lower and upper limits for the y-axis
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title : str, optional
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If specified, provides a title for the plot
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c : color, default='#30a2da'
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Color for the bars
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**kwargs : keywords, optional
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extra keyword arguments passed to ax.bar()
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"""
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ax = plt.gca()
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if x is None:
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x = np.arange(len(pos))
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ax.bar(x, pos, color=c, **kwargs)
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if ylim:
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plt.ylim(ylim)
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plt.xticks(np.asarray(x)+0.4, x)
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if title is not None:
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plt.title(title)
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def plot_belief_vs_prior(belief, prior, **kwargs):
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""" plots two discrete probability distributions side by side, with
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titles "belief" and "prior"
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"""
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plt.subplot(121)
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bar_plot(belief, title='belief', **kwargs)
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plt.subplot(122)
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bar_plot(prior, title='prior', **kwargs)
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def plot_prior_vs_posterior(prior, posterior, reverse=False, **kwargs):
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""" plots two discrete probability distributions side by side, with
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titles "prior" and "posterior"
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"""
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if reverse:
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plt.subplot(121)
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bar_plot(posterior, title='posterior', **kwargs)
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plt.subplot(122)
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bar_plot(prior, title='prior', **kwargs)
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else:
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plt.subplot(121)
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bar_plot(prior, title='prior', **kwargs)
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plt.subplot(122)
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bar_plot(posterior, title='posterior', **kwargs)
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def set_labels(title=None, x=None, y=None):
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""" helps make code in book shorter. Optional set title, xlabel and ylabel
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"""
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if x is not None:
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plt.xlabel(x)
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if y is not None:
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plt.ylabel(y)
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if title is not None:
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plt.title(title)
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def set_limits(x, y):
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""" helper function to make code in book shorter. Set the limits for the x
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and y axis.
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"""
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plt.gca().set_xlim(x)
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plt.gca().set_ylim(y)
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def plot_predictions(p, rng=None):
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if rng is None:
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rng = range(len(p))
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plt.scatter(rng, p, marker='v', s=40, edgecolor='r',
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facecolor='None', lw=2, label='prediction')
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def plot_kf_output(xs, filter_xs, zs, title=None, aspect_equal=True):
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|
plot_filter(filter_xs[:, 0])
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plot_track(xs[:, 0])
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|
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|
if zs is not None:
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|
plot_measurements(zs)
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|
show_legend()
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|
set_labels(title=title, y='meters', x='time (sec)')
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|
if aspect_equal:
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|
plt.gca().set_aspect('equal')
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|
plt.xlim((-1, len(xs)))
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|
plt.show()
|
|
|
|
|
|
def plot_measurements(xs, ys=None, color='k', lw=2, label='Measurements',
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|
lines=False, **kwargs):
|
|
""" Helper function to give a consistant way to display
|
|
measurements in the book.
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|
"""
|
|
|
|
plt.autoscale(tight=True)
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|
if lines:
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|
if ys is not None:
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|
return plt.plot(xs, ys, color=color, lw=lw, ls='--', label=label, **kwargs)
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|
else:
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|
return plt.plot(xs, color=color, lw=lw, ls='--', label=label, **kwargs)
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else:
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|
if ys is not None:
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|
return plt.scatter(xs, ys, edgecolor=color, facecolor='none',
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|
lw=2, label=label, **kwargs),
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|
else:
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|
return plt.scatter(range(len(xs)), xs, edgecolor=color, facecolor='none',
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|
lw=2, label=label, **kwargs),
|
|
|
|
|
|
def plot_residual_limits(Ps, stds=1.):
|
|
""" plots standand deviation given in Ps as a yellow shaded region. One std
|
|
by default, use stds for a different choice (e.g. stds=3 for 3 standard
|
|
deviations.
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|
"""
|
|
|
|
std = np.sqrt(Ps) * stds
|
|
|
|
plt.plot(-std, color='k', ls=':', lw=2)
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|
plt.plot(std, color='k', ls=':', lw=2)
|
|
plt.fill_between(range(len(std)), -std, std,
|
|
facecolor='#ffff00', alpha=0.3)
|
|
|
|
|
|
def plot_track(xs, ys=None, label='Track', c='k', lw=2, **kwargs):
|
|
if ys is not None:
|
|
return plt.plot(xs, ys, color=c, lw=lw, ls=':', label=label, **kwargs)
|
|
else:
|
|
return plt.plot(xs, color=c, lw=lw, ls=':', label=label, **kwargs)
|
|
|
|
|
|
def plot_filter(xs, ys=None, c='#013afe', label='Filter', var=None, **kwargs):
|
|
#def plot_filter(xs, ys=None, c='#6d904f', label='Filter', vars=None, **kwargs):
|
|
|
|
|
|
if ys is None:
|
|
ys = xs
|
|
xs = range(len(ys))
|
|
|
|
plt.plot(xs, ys, color=c, label=label, **kwargs)
|
|
|
|
if var is None:
|
|
return
|
|
|
|
var = np.asarray(var)
|
|
|
|
std = np.sqrt(var)
|
|
std_top = ys+std
|
|
std_btm = ys-std
|
|
|
|
plt.plot(xs, ys+std, linestyle=':', color='k', lw=2)
|
|
plt.plot(xs, ys-std, linestyle=':', color='k', lw=2)
|
|
plt.fill_between(xs, std_btm, std_top,
|
|
facecolor='yellow', alpha=0.2)
|
|
|
|
|
|
|
|
|
|
def _blob(x, y, area, colour):
|
|
"""
|
|
Draws a square-shaped blob with the given area (< 1) at
|
|
the given coordinates.
|
|
"""
|
|
hs = np.sqrt(area) / 2
|
|
xcorners = np.array([x - hs, x + hs, x + hs, x - hs])
|
|
ycorners = np.array([y - hs, y - hs, y + hs, y + hs])
|
|
plt.fill(xcorners, ycorners, colour, edgecolor=colour)
|
|
|
|
def hinton(W, maxweight=None):
|
|
"""
|
|
Draws a Hinton diagram for visualizing a weight matrix.
|
|
Temporarily disables matplotlib interactive mode if it is on,
|
|
otherwise this takes forever.
|
|
"""
|
|
reenable = False
|
|
if plt.isinteractive():
|
|
plt.ioff()
|
|
|
|
plt.clf()
|
|
height, width = W.shape
|
|
if not maxweight:
|
|
maxweight = 2**np.ceil(np.log(np.max(np.abs(W)))/np.log(2))
|
|
|
|
plt.fill(np.array([0, width, width, 0]),
|
|
np.array([0, 0, height, height]),
|
|
'gray')
|
|
|
|
plt.axis('off')
|
|
plt.axis('equal')
|
|
for x in range(width):
|
|
for y in range(height):
|
|
_x = x+1
|
|
_y = y+1
|
|
w = W[y, x]
|
|
if w > 0:
|
|
_blob(_x - 0.5,
|
|
height - _y + 0.5,
|
|
min(1, w/maxweight),
|
|
'white')
|
|
elif w < 0:
|
|
_blob(_x - 0.5,
|
|
height - _y + 0.5,
|
|
min(1, -w/maxweight),
|
|
'black')
|
|
if reenable:
|
|
plt.ion()
|
|
|
|
|
|
if __name__ == "__main__":
|
|
|
|
plot_errorbar1()
|
|
plot_errorbar2()
|
|
plot_errorbar3()
|
|
plot_hypothesis1()
|
|
plot_hypothesis2()
|
|
plot_hypothesis3()
|
|
plot_hypothesis4()
|
|
plot_hypothesis5()
|
|
plot_estimate_chart_1()
|
|
plot_estimate_chart_2()
|
|
plot_estimate_chart_3()
|
|
create_predict_update_chart()
|
|
show_residual_chart()
|
|
show_residual_chart(True, True)
|
|
plt.close('all')
|
|
|
|
'''p = [0.2245871, 0.06288015, 0.06109133, 0.0581008, 0.09334062, 0.2245871,
|
|
0.06288015, 0.06109133, 0.0581008, 0.09334062]*2
|
|
bar_plot(p)
|
|
plot_measurements(p)''' |