r""".. _sphara_analysis_eeg:


Spatial SPHARA analysis of EEG data
===================================

.. topic:: Section contents

   This tutorial shows exemplarily the spatial SPHARA analysis of
   256-channel EEG data. The FEM discretization of the
   Laplace–Beltrami operator is employed to calculate the SPHARA basic
   functions that are used for the SPHARA decomposition.


Introduction
------------

As explained in :ref:`introduction` and in the tutorial
:ref:`sphara_basis_eeg` a spatial Fourier basis for a arbitrary sensor
setup can be determined as solution of the Laplace's eigenvalue
problem dicretized for the considered sensor setup

.. math::

   \mathbf{L} \boldsymbol{\varphi}_i = \lambda_i \boldsymbol{\varphi}_i\,,

with the discrete :term:`Laplace–Beltrami operator`
:math:`\mathbf{L}` in matrix notation, eigenvectors
:math:`\boldsymbol{\varphi}_i` containing the spatial harmonic
functions and eigenvalues :math:`\lambda_i \ge 0`. For a FEM
discretization of the Laplace–Beltrami operator the quantities
:math:`\sqrt{\lambda_i}` can be interpreted as spatial angular
frequencies; see :ref:`introduction`.

The spatial Fourier basis determined in this way can be used for the
spatial Fourier analysis of data recorded with the considered sensor
setup.

For the anaylsis of discrete data defined on the vertices of the
triangular mesh - the SPHARA transform - the inner product is used
(transformation from spatial domain to spatial frequency domain). For
an analysis using eigenvectors computed by the FEM approach, the inner
product that assures the :math:`\boldsymbol{B}`-orthogonality needs to
be applied.

For the reverse transformation, the discrete data are synthesized
using the linear combination of the SPHARA coefficients and the
corresponding SPHARA basis functions. More detailed information can be
found in the section :ref:`sphara_analysis_synthesis` and in
:cite:`graichen15`.

"""

######################################################################
# At the beginning we import three modules of the SpharaPy package as
# well as several other packages and single functions of packages.

# Code source: Uwe Graichen
# License: BSD 3 clause

# import modules from spharapy package
# import additional modules used in this tutorial
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.mplot3d import Axes3D  # noqa: F401  (registers 3D)

import spharapy.datasets as sd
import spharapy.spharatransform as st
import spharapy.trimesh as tm

######################################################################
# Import the spatial configuration of the EEG sensors and the SEP data
# --------------------------------------------------------------------
# In this tutorial we will apply the SPHARA analysis to SEP data of a
# single subject recorded with a 256 channel EEG system with
# equidistant layout. The data set is one of the example data sets
# contained in the SpharaPy toolbox.

# loading the 256 channel EEG dataset from spharapy sample datasets
mesh_in = sd.load_eeg_256_channel_study()

######################################################################
# The dataset includes lists of vertices, triangles, and sensor
# labels, as well as EEG data from previously performed experiment
# addressing the cortical activation related to somatosensory-evoked
# potentials (SEP).

print(mesh_in.keys())

######################################################################
# The triangulation of the EEG sensor setup consists of 256 vertices
# and 480 triangles. The EEG data consists of 256 channels and 369
# time samples, 50 ms before to 130 ms after stimulation. The sampling
# frequency is 2048 Hz.

vertlist = np.array(mesh_in["vertlist"])
trilist = np.array(mesh_in["trilist"])
eegdata = np.array(mesh_in["eegdata"])
print("vertices = ", vertlist.shape)
print("triangles = ", trilist.shape)
print("eegdata = ", eegdata.shape)

######################################################################

fig = plt.figure()
fig.subplots_adjust(left=0.02, right=0.98, top=0.98, bottom=0.02)
ax = fig.add_subplot(111, projection="3d")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
ax.set_title("The triangulated EEG sensor setup")
ax.view_init(elev=20.0, azim=80.0)
ax.set_aspect("auto")
ax.plot_trisurf(
    vertlist[:, 0],
    vertlist[:, 1],
    vertlist[:, 2],
    triangles=trilist,
    color="lightblue",
    edgecolor="black",
    linewidth=0.5,
    shade=True,
)
plt.show()

######################################################################

x = np.arange(-50, 130, 1 / 2.048)
figeeg = plt.figure()
axeeg = figeeg.gca()
axeeg.plot(x, eegdata[:, :].transpose())
axeeg.set_xlabel("t/ms")
axeeg.set_ylabel("V/µV")
axeeg.set_title("SEP data")
axeeg.set_ylim(-3.5, 3.5)
axeeg.set_xlim(-50, 130)
axeeg.grid(True)
plt.show()


######################################################################
# Create a SpharaPy TriMesh instance
# ----------------------------------
#
# In the next step we create an instance of the class
# :class:`spharapy.trimesh.TriMesh` from the list of vertices and
# triangles.

# create an instance of the TriMesh class
mesh_eeg = tm.TriMesh(trilist, vertlist)


######################################################################
# SPHARA transform using FEM discretisation
# -----------------------------------------
# Create a SpharaPy SpharaTransform instance
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# In the next step of the tutorial we determine an instance of the
# class SpharaTransform, which is used to execute the
# transformation. For the determination of the SPHARA basis we use a
# Laplace–Beltrami operator, which is discretized by the FEM
# approach.

sphara_transform_fem = st.SpharaTransform(mesh_eeg, "fem")
basis_functions_fem, natural_frequencies_fem = sphara_transform_fem.basis()

######################################################################
# Visualization the basis functions
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# The first 15 spatially low-frequency SPHARA basis functions of the
# basis used for the transform are shown below, starting with DC
# at the top left.

figsb1, axes1 = plt.subplots(nrows=5, ncols=3, figsize=(8, 12), subplot_kw={"projection": "3d"})
for i in range(np.size(axes1)):
    colors = np.mean(basis_functions_fem[trilist, i + 0], axis=1)
    ax = axes1.flat[i]
    ax.set_xlabel("x")
    ax.set_ylabel("y")
    ax.set_zlabel("z")
    ax.view_init(elev=60.0, azim=80.0)
    ax.set_aspect("auto")
    trisurfplot = ax.plot_trisurf(
        vertlist[:, 0],
        vertlist[:, 1],
        vertlist[:, 2],
        triangles=trilist,
        cmap=plt.cm.bwr,
        edgecolor="white",
        linewidth=0.0,
    )
    trisurfplot.set_array(colors)
    trisurfplot.autoscale()
    trisurfplot.set_clim(-0.01, 0.01)

cbar = figsb1.colorbar(
    trisurfplot,
    ax=axes1.ravel().tolist(),
    shrink=0.85,
    orientation="horizontal",
    fraction=0.05,
    pad=0.05,
    anchor=(0.5, -4.5),
)

plt.subplots_adjust(left=0.0, right=1.0, bottom=0.08, top=1.0)
plt.show()


######################################################################
# SPHARA transform of the EEG data
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
#
# In the final step we perform the SPHARA transformation of the EEG
# data. As a result, a butterfly plot of all channels of the EEG is
# compared to the visualization of the power contributions of the
# first 40 SPHARA basis functions. Only the first 40 out of 256 basis
# functions are used for the visualization, since the power
# contribution of the higher basis functions is very low.

# perform the SPHARA transform
sphara_trans_eegdata = sphara_transform_fem.analysis(eegdata.transpose())

# 40 low-frequency basis functions are displayed
ysel = 40
figsteeg, (axsteeg1, axsteeg2) = plt.subplots(nrows=2)

y = np.arange(0, ysel)
x = np.arange(-50, 130, 1 / 2.048)

axsteeg1.plot(x, eegdata[:, :].transpose())
axsteeg1.set_ylabel("V/µV")
axsteeg1.set_title("EEG data, 256 channels")
axsteeg1.set_ylim(-2.5, 2.5)
axsteeg1.set_xlim(-50, 130)
axsteeg1.grid(True)

pcm = axsteeg2.pcolormesh(x, y, np.square(np.abs(sphara_trans_eegdata.transpose()[0:ysel, :])))
axsteeg2.set_xlabel("t/ms")
axsteeg2.set_ylabel("# BF")
axsteeg2.set_title("Power contribution of SPHARA basis functions")
axsteeg2.grid(True)
figsteeg.colorbar(
    pcm, ax=[axsteeg1, axsteeg2], shrink=0.45, anchor=(0.85, 0.0), label="power / a.u."
)

plt.subplots_adjust(left=0.1, right=0.85, bottom=0.1, top=0.95, hspace=0.35)
plt.show()
# sphinx_gallery_thumbnail_number = 4