# pylint: disable=wrong-import-position,wrong-import-order,ungrouped-imports,invalid-name
r"""
.. _apps-points-tutorial:
Point processes
===============
*Technical details are available in the API documentation:* :doc:`code/api/strawberryfields.apps.points`
This section shows how to generate GBS point process samples and use them to detect outlier
points in a data set. Point processes are models for generating random point patterns and can be
useful in machine learning, providing a source of randomness with
preference towards both diversity [[#kulesza2012determinantal]_] and similarity in data. GBS
devices can be programmed to operate as special types of point processes that generate clustered
random point patterns [[#jahangiri2019point]_].
The probability of generating a specific pattern of points in GBS point processes depends on
matrix functions of a kernel matrix :math:`K` that describes the similarity between the points.
Matrix functions that appear in GBS point processes are typically
`permanents `__ and
`hafnians `__. Here we use
the permanental point process, in which the probability of observing a pattern of points :math:`S`
depends on the permanent of their corresponding kernel submatrix :math:`K_S` as
[[#jahangiri2019point]_]:
.. math::
\mathcal{P}(S) = \frac{1}{\alpha(S)}\text{per}(K_S),
where :math:`\alpha` is a normalization function that depends on :math:`S` and the average number
of points. Let's look at a simple example to better understand the permanental point process.
"""
##############################################################################
# We first import the modules we need. Note that the :mod:`~strawberryfields.apps.points` module has most of
# the core functionalities exploring point processes.
import numpy as np
import plotly
from sklearn.datasets import make_blobs
from strawberryfields.apps import points, plot
##############################################################################
# We define a space where the GBS point process patterns are generated. This
# space is referred to as the state space and is defined by a set of points. The
# point process selects a subset of these points in each sample. Here we create
# a 20 :math:`\times` 20 square grid of points.
R = np.array([(i, j) for i in range(20) for j in range(20)])
##############################################################################
# The rows of R are the coordinates of the points.
#
# Next step is to create the kernel matrix for the points of this discrete space. We call the
# :func:`~strawberryfields.apps.points.rbf_kernel` function which uses the *radial basis function*
# (RBF) kernel defined as:
#
# .. math::
# K_{i,j} = e^{-\|\bf{r}_i-\bf{r}_j\|^2/2\sigma^2},
#
# where :math:`\bf{r}_i` are the coordinates of point :math:`i` and :math:`\sigma` is a kernel
# parameter that determines the scale of the kernel.
#
# In the RBF kernel, points that are much further than a distance :math:`\sigma` from each other
# lead to small entries of the kernel matrix, whereas points much closer than :math:`\sigma`
# generate large entries. Now consider a specific point pattern in which all points
# are close to each other, which simply means that their matrix elements have larger entries. The
# permanent of a matrix is a sum over the product of some matrix entries. Therefore,
# the submatrix that corresponds to those points has a large permanent and the probability of
# observing them in a sample is larger.
#
# For kernel matrices that are positive-semidefinite, such as the RBF kernel, there exist efficient
# quantum-inspired classical algorithms for permanental point process sampling
# [[#jahangiri2019point]_]. In this tutorial we use positive-semidefinite kernels and the
# quantum-inspired classical algorithm.
#
# Let's construct the RBF kernel with the parameter :math:`\sigma` set to 2.5.
K = points.rbf_kernel(R, 2.5)
##############################################################################
# We generate 10 samples with an average number of 50 points per sample by calling
# the :func:`~strawberryfields.apps.points.sample` function of the :mod:`~strawberryfields.apps.points` module.
samples = points.sample(K, 50.0, 10)
##############################################################################
# We visualize the first sample by using the :func:`~strawberryfields.apps.plot.points` function of
# the :mod:`~strawberryfields.apps.plot` module. The point patterns generated by the permanental point process
# usually have a higher degree of clustering compared to a uniformly random pattern.
plot.points(R, samples[0], point_size=10)
##############################################################################
# Outlier Detection
# -----------------
#
# When the distribution of points in a given space is inhomogeneous, GBS point processes
# sample points from the dense regions with higher probability. This feature of the GBS point
# processes can be used to detect outlier points in a data set. In this example, we create two
# dense clusters and place them in a two-dimensional space containing some randomly distributed
# points in the background. We consider the random background points as outliers to the clustered
# points and show that the permanental point process selects points from the dense clusters with
# a higher probability.
#
# We first create the data points. The clusters have 50 points each and the points have a
# standard deviation of 0.3. The clusters are centered at :math:`[x = 2, y = 2]` and :math:`[x = 4,
# y = 4]`, respectively. We also add 25 randomly generated points to the data set.
clusters = make_blobs(n_samples=100, centers=[[2, 2], [4, 4]], cluster_std=0.3)[0]
noise = np.random.rand(25, 2) * 6.0
R = np.concatenate((clusters, noise))
##############################################################################
# Then construct the kernel matrix and generate 10000 samples.
K = points.rbf_kernel(R, 1.0)
samples = points.sample(K, 10.0, 10000)
##############################################################################
# We obtain the indices of 100 points that appear most frequently in the permanental point
# process samples and visualize them. The majority of the commonly appearing points belong
# to the clusters and the points that do not appear frequently are the outlier points. Note that
# some of the background points might overlap with the clusters.
gbs_frequent_points = np.argsort(np.sum(samples, axis=0))[-100:]
plot.points(
R, [1 if i in gbs_frequent_points else 0 for i in range(len(samples[0]))], point_size=10
)
##############################################################################
# The two-dimensional examples considered here can be easily extended to higher dimensions. The
# GBS point processes retain their clustering property in higher dimensions but visual inspection
# of this clustering feature might not be very straightforward.
#
# GBS point processes can potentially be used in other applications such as clustering data points
# and finding correlations in time series data. Can you design your own example for using GBS point
# processes in a new application?
#
# References
# ----------
#
# .. [#kulesza2012determinantal]
#
# Alex Kulesza, Ben Taskar, and others. Determinantal point processes for machine learning.
# Foundations and Trends® in Machine Learning, 5(2–3):123–286, 2012.
#
# .. [#jahangiri2019point]
#
# Soran Jahangiri, Juan Miguel Arrazola, Nicolás Quesada, and Nathan Killoran. Point processes with
# gaussian boson sampling. 2019. arXiv:1906.11972.