#!/usr/bin/env python
# coding: utf-8
"""
====================================
Particle Filter Resamplers: Tutorial
====================================
"""
import numpy as np
# %%
# Introduction
# ------------
# Stone Soup comes with a number of resamplers, for the Particle Filter, that can be
# used straight out of the box. This example explains each of the resamplers and compares the
# results of using each one.
#
#
# **Resamplers currently available in Stone Soup:**
#
# - :class:`~.SystematicResampler`
# - :class:`~.MultinomialResampler`
# - :class:`~.StratifiedResampler`
# - :class:`~.ResidualResampler`
# - :class:`~.ESSResampler` (Effective Sample Size Resampler)
#
# The last two resamplers (Residual and ESS) are preprocessing methods that require the use of
# another resampler.
# %%
# Plotter for this notebook
# ^^^^^^^^^^^^^^^^^^^^^^^^^
import matplotlib.pyplot as plt
def plot(normalised_weights, u_j=None, stratified=False, residual=False):
nparts = len(normalised_weights)
floors = np.floor(normalised_weights * nparts)
# Plot
fig, ax = plt.subplots()
fig.set_size_inches(10, 1.5)
plt.xlim([0, 1])
for i, particle_weight in enumerate(normalised_weights):
l = 0
for j in range(i):
l += normalised_weights[j]
ax.barh(['CDF'], [particle_weight], left=l)
if residual is not False:
if floors[i] != 0:
ax.barh(['CDF'], [(1 / nparts) * floors[i]], left=l, color='White')
if u_j is not None:
plt.plot([u_j, u_j], [-0.4, 0.4], 'k-', lw=4)
if stratified is not False:
plt.plot([s_lb, s_lb], [-0.4, 0.4], 'w:', lw=2)
# %%
# Cumulative Distribution Function
# --------------------------------
#
# The Systematic, Multinomial, and Stratified resamplers all use a cumulative distribution
# function (CDF) to randomly pick points from. The CDF is calculated from the cumulative sum
# of the normalised weights of the existing particles.
#
# **Example**
#
# Given the following set of particles and their weights:
#
# .. list-table:: Particle Weights
# :widths: 25 25 50
# :header-rows: 1
#
# * - Particle
# - Weight
# - Normalised Weight
# * - 1
# - 1
# - 0.1
# * - 2
# - 5
# - 0.5
# * - 3
# - 1.5
# - 0.15
# * - 4
# - 0.5
# - 0.05
# * - 5
# - 2
# - 0.2
#
# We would calculate the following CDF:
particle_weights = [1, 5, 1.5, 0.5, 2]
n_particles = len(particle_weights)
normalised_weights = np.array([particle/sum(particle_weights) for particle in particle_weights])
plot(normalised_weights)
# %%
# Each of the resamplers then use a different method to pick points in the CDF. For example,
# assume the following points are picked: 0.15, 0.41, 0.57, 0.63, and 0.89. These points are
# shown on the CDF below.
u_j = [0.15, 0.41, 0.57, 0.63, 0.89]
plot(normalised_weights, u_j)
# %%
# As shown above, the points 0.15, 0.41, and 0.57 all fall within the section of the CDF
# corresponding to particle 2, while the points 0.63 and 0.89 fall within the sections of
# the CDF corresponding to particles 3 and 5, respectively.
#
# Hence, for this example, the number of times each particle is resampled is as follows:
#
# .. list-table:: Particle Weights
# :widths: 25 25 50
# :header-rows: 1
#
# * - Particle
# - No. of times resampled
# - Weight of new resampled particles
# * - 1
# - 0
# - 0.2
# * - 2
# - 3
# - 0.2
# * - 3
# - 1
# - 0.2
# * - 4
# - 0
# - 0.2
# * - 5
# - 1
# - 0.2
#
# As shown in the table, the resampler assigns a new weight to each sample - by default, each of
# the resamplers included in Stone Soup give all particles an equal weight of :math:`1/N` where
# :math:`N` = no. of resampled particles.
# %%
# Resamplers
# ----------
# %%
# Multinomial Resampler
# ^^^^^^^^^^^^^^^^^^^^^
#
# The :class:`~.MultinomialResampler` calculates :math:`N` independent random numbers from the
# uniform distribution :math:`u \sim U(0,1]`, where :math:`N` is the target number of particles to
# be resampled. In most cases, we use :math:`N = M`, where :math:`M` is the number of existing
# particles to be resampled. However, the Multinomial resampler can upsample (:math:`N > M`) or
# downsample (:math:`N < M`) to a value :math:`N \neq M \in \mathbb{N}`. The Multinomial resampler
# has a computational complexity of :math:`O(MN)` where :math:`N` and :math:`M` are the number of
# resampled and existing particles respectively.
#
# An example of how the Multinomial resampler picks points from the CDF is shown below. Black
# lines represent the chosen points.
import numpy as np
# Pick N random points in the uniform distribution u~U(0,1]
u_j = np.random.rand(5)
plot(normalised_weights, u_j)
# %%
# Systematic Resampler
# ^^^^^^^^^^^^^^^^^^^^
#
# Unlike the Multinomial resampler, the :class:`~.SystematicResampler` doesn't calculate all points
# independently. Instead, a single random starting point in the range :math:`[0,1/N]` is chosen. :math:`N` points are then
# calculated at equidistant intervals along the CDF, so that there is a gap of :math:`1/N`
# between any two consecutive points. The Systematic resampler has a computational complexity
# of :math:`O(N)` where :math:`N` is the number of resampled particles.
#
# An example of how the Systematic resampler picks points from the CDF is shown below. Black
# lines represent the chosen points.
# Pick a starting point
s = np.random.uniform(0, 1/n_particles)
# Calculate N equidistant points from the starting point
u_j = s + (1 / n_particles) * np.arange(n_particles)
plot(normalised_weights, u_j)
# %%
# Stratified Resampler
# ^^^^^^^^^^^^^^^^^^^^
#
# The :class:`~.StratifiedResampler` splits the whole CDF into :math:`N` evenly sized strata
# (subpopulations). A random point is then chosen, independently, from each stratum. This results
# in the gap between any two consecutive points being in the range :math:`(0, 2/N]`. The
# Stratified resampler has a computational complexity of :math:`O(N)` where :math:`N` is the
# number of resampled particles.
#
# An example of how the Stratified resampler picks points from the CDF is shown below. Black
# lines represent the chosen points, with white dashed lines representing the strata boundaries.
# Calculate lower bound of each stratum
s_lb = np.arange(n_particles) * (1 / n_particles)
# Independently pick a point from each stratum
u_j = np.random.uniform(s_lb, s_lb + (1 / n_particles))
plot(normalised_weights, u_j, s_lb)
# %%
# Preprocessing methods
# ---------------------
# %%
# Residual Resampler
# ^^^^^^^^^^^^^^^^^^
#
# The :class:`~.ResidualResampler` consists of two stages.
#
# **Stage 1**
#
# The first stage determines which particles have weight :math:`w^{i} \geq 1/N`, where
# :math:`i \in 1, ..., N` denotes each particle. Each of these particles is then resampled
# :math:`N^{i}_{j} = floor(Nw^{i}_{j})` times, where :math:`j \in 1, 2` denotes the stage. Hence,
# :math:`N_1 = \sum_{i=1}^{N}N^i_1` represents the number of particles that are sampled in stage 1.
# As these weights have been represented in the resampled set of particles, we are only interested
# in the residual weights, left after the floor weights (:math:`N^{i}_{1}`) have been subtracted.
#
#
# Reconsider our example with 5 particles, with weights shown by the plot below.
plot(normalised_weights)
# %%
# As we're interested in particles of weight :math:`w^{i} \geq 1/N`, in this example we're looking
# for particles with :math:`w^{i} \geq 1/5`. Hence, Particle 2 (orange, weight = 0.5) and Particle
# 5 (purple, weight = 0.2). We resample these particles :math:`N^{i}_{j} = floor(Nw^{i}_{j})`
# times:
#
# - Particle 2 gets resampled :math:`N^{2}_{1} = floor(Nw^{2}_{1}) = floor(5 * 0.5) = 2` times.
# - Particle 5 gets resampled :math:`N^{5}_{1} = floor(Nw^{5}_{1}) = floor(5 * 0.2) = 1` times.
# - Particles 1, 3 and 4 do not get resampled as their weights are :math:`\leq 1/5`.
#
#
# A total of 3 particles were sampled from stage 1 (2x Particle 2, 1x Particle 5), hence
# :math:`N_1 = \sum_{i=1}^{N}N^i_1 = 3`
#
#
# Original CDF with floor weights removed:
plot(normalised_weights, residual=True)
# %%
# New CDF:
floors = np.floor(normalised_weights * n_particles)
residuals = (normalised_weights - floors/n_particles)
plot(residuals)
# %%
# The CDF of the residual weights is carried on to stage 2.
#
#
# **Stage 2**
#
#
# We now look to resample the remaining :math:`N_2 = N - N_1` particles during stage 2. (Reminder:
# N is the total number of particles we want to sample, while :math:`N_1` is the number of
# particles we have already resampled from stage 1). The residual weights from each particle are
# carried over from stage 1. These residual weights are normalised, and used to calculate a CDF,
# which is shown below.
#
#
# Normalised residual weight CDF:
normalised_residual_weights = residuals*(1/sum(residuals))
plot(normalised_residual_weights)
# %%
# We then use either the :class:`~.MultinomialResampler`, :class:`~.StratifiedResampler`, or
# :class:`~.SystematicResampler` to sample the remaining :math:`N_2` particles using the CDF.
#
#
# Continuing with our example, we want to resample the remaining :math:`N_2 = N - N_1 = 5 - 3 = 2`
# particles in stage 2. Here we will choose to use the :class:`~.MultinomialResampler` (we could
# also use :class:`~.StratifiedResampler` or :class:`~.SystematicResampler`) to sample
# the :math:`N_2 = 2` particles from the normalised residual weight CDF - more detail on how this
# is done can be found in the Multinomial Resampler section above.
u_j = np.random.rand(2)
plot(normalised_residual_weights, u_j)
# %%
#
# This method reduces the number of particles that are sampled through the more computationally
# expensive methods seen above. The Residual resampler also guarantees that any particle of weight
# greater than :math:`1/N` will be represented in the set of resampled particles.
#
# When using the Residual resampler in Stone Soup, the Resampler requires a property
# 'residual_resampler'. This property defines which resampler method to use for resampling the
# residuals in stage 2.
# The variable must be a string value from the following: 'multinomial', 'systematic',
# 'stratified'. If no `residual_method` variable is provided, the multinomial method will be used
# by default.
# %%
# ESS Resampler
# ^^^^^^^^^^^^^
#
# The :class:`~.ESSResampler` (Effective Sample Size) is a wrapper around another resampler. It
# performs a check at each time step to determine whether it is necessary to resample the
# particles. Resampling is only performed at a given time step if a defined criterion is met.
# Kish's ESS criterion is used, which states that:
#
# .. math::
# ESS = \left(\sum_{i=1}^{N} (W_{n}^i)^2\right)^{-1}
#
# is less than or equal to the threshold, a float which is set by default to :math:`\frac{N}{2}`.
# %%
# Example in Stone Soup
# ----------------------
#
# Generate some particles
# ^^^^^^^^^^^^^^^^^^^^^^^
#
# An example of resampling particles using the Stone Soup resamplers. We generate some particles
# using the :class:`~.Particle` class. In this example, we give every particle an equal
# weight - each particle will have the same likelihood of being resampled. The resample function
# returns a :class:`~.ParticleState`. The state_vector property of which contains a list of
# particles to be resampled.
from stonesoup.types.particle import Particle
particles = [Particle(np.array([[i]]), weight=1/5) for i in range(5)]
# %%
# Simple Example
# """"""""""""""
#
# In this example, we use the :class:`~.MultinomialResampler` to resample the particles generated
# above. We simply define the Resampler and call its `resample` method.
#
# There are situations where we may want to resample to a different number of particles - for
# example if you want more granular information, or want to be more computationally efficient.
# Excluding the :class:`~.ResidualResampler`, all Stone Soup resamplers can up- or down-sample as
# shown below.
from stonesoup.resampler.particle import MultinomialResampler
# Resampler
resampler = MultinomialResampler()
# Resample particles
resampled_particles = resampler.resample(particles)
print("---- State vector of resampled particles ----")
print(resampled_particles.state_vector)
print("------ Weights of resampled particles -------")
print(resampled_particles.weight)
# Repeat while upsampling particles
upsampled_particles = resampler.resample(particles, nparts=10)
print("---- State vector of upsampled particles ----")
print(upsampled_particles.state_vector)
print("------ Weights of upsampled particles -------")
print(upsampled_particles.weight)
# %%
# Here we can see which particles were chosen to be resampled and the weights of the new
# particles.
# %%
# Example using ESS and ResidualResampler
# """""""""""""""""""""""""""""""""""""""
#
# In this example, we use both the Effective Sample Size method, and the residual resampler, using
# the systematic method to resample the residuals.
from stonesoup.resampler.particle import ESSResampler, ResidualResampler
# Define Resampler
subresampler = ResidualResampler(residual_method='systematic')
resampler = ESSResampler(resampler=subresampler)
# Resample Particles
resampler.resample(particles)