Component analysis

An introduction to component analysis in phasor space.

Import required modules, functions, and classes:

import matplotlib.animation
import numpy
from matplotlib import pyplot

from phasorpy.component import (
    phasor_component_fit,
    phasor_component_fraction,
    phasor_component_graphical,
)
from phasorpy.lifetime import phasor_from_lifetime
from phasorpy.plot import PhasorPlot, plot_histograms

rng = numpy.random.default_rng(42)  # initialize random number generator

component_style = {
    'linestyle': '-',
    'marker': 'o',
    'color': 'tab:blue',
    'fontsize': 14,
}

Fractions of two components

The phasor coordinates of a combination of two lifetime components lie on the line between the two components. For example, a mixture of:

• Component A: 1.0 ns lifetime, 60% contribution

• Component B: 8.0 ns lifetime, 40% contribution

frequency = 80.0  # MHz
component_lifetimes = [1.0, 8.0]  # ns
component_fractions = [0.6, 0.4]

component_real, component_imag = phasor_from_lifetime(
    frequency, component_lifetimes
)

plot = PhasorPlot(frequency=frequency, title='Fractions of two components')
plot.components(
    component_real,
    component_imag,
    component_fractions,
    labels=['A', 'B'],
    **component_style,
)
plot.show()

If the location of both components is known, their contributions (fractions) to a phasor point that lies on the line between the components can be calculated:

real, imag = phasor_from_lifetime(
    frequency, component_lifetimes, component_fractions
)

fraction_of_first_component = phasor_component_fraction(
    real, imag, component_real, component_imag
)

assert numpy.isclose(fraction_of_first_component, component_fractions[0])

Distribution of fractions of two components

A distribution of phasor coordinates can represent varying contributions from two components with known phasor coordinates:

real, imag = rng.multivariate_normal(
    [real, imag], [[5e-3, 1e-3], [1e-3, 1e-3]], (100, 100)
).T

plot = PhasorPlot(
    frequency=frequency, title='Distribution of fractions of two components'
)
plot.hist2d(real, imag, cmap='Greys')
plot.components(
    component_real, component_imag, labels=['A', 'B'], **component_style
)
plot.show()

When the phasor coordinates of two contributing components are known, their fractional contributions to phasor coordinates can be calculated by projecting the phasor coordinates onto the line connecting the components. Fractions are calculated using phasorpy.component.phasor_component_fraction() and plotted as histograms:

fraction_of_first_component = phasor_component_fraction(
    real, imag, component_real, component_imag
)

plot_histograms(
    fraction_of_first_component,
    1.0 - fraction_of_first_component,
    range=(0, 1),
    bins=100,
    alpha=0.66,
    labels=['A', 'B'],
    xlabel='Fraction',
    ylabel='Count',
    title='Histograms of fractions of two components',
)

Multi-component fit

When the phasor coordinates of multiple contributing components are known, their fractional contributions to phasor coordinates can be obtained by solving a linear system of equations, using multiple harmonics if necessary.

Fractions of two components are fitted using phasorpy.component.phasor_component_fit() and plotted as histograms:

fraction1, fraction2 = phasor_component_fit(
    numpy.ones_like(real), real, imag, component_real, component_imag
)

plot_histograms(
    fraction1,
    fraction2,
    range=(0, 1),
    bins=100,
    alpha=0.66,
    labels=['A', 'B'],
    xlabel='Fraction',
    ylabel='Count',
    title='Histograms of fitted fractions of multiple components',
)

Up to three components can be fitted to single-harmonic phasor coordinates. The Multi-component fit tutorial demonstrates how to fit five components using two harmonics.

Graphical analysis of two components

The phasorpy.component.phasor_component_graphical() function for two components counts the number of phasor coordinates that fall within a radius at given fractions along the line between the components. The plot below shows counts versus fraction, which can be compared to the previous histogram:

radius = 0.025
fractions = numpy.linspace(0.0, 1.0, 20)

counts = phasor_component_graphical(
    real,
    imag,
    component_real,
    component_imag,
    fractions=fractions,
    radius=radius,
)

fig, ax = pyplot.subplots()
ax.plot(fractions, counts, '-', label='A vs B')
ax.set_title('Graphical analysis of two components')
ax.set_xlabel('Fraction')
ax.set_ylabel('Count')
ax.legend()
pyplot.show()

Graphical analysis of three components

The graphical method similarly applies to the contributions of three components:

component_lifetimes = [1.0, 4.0, 15.0]
component_real, component_imag = phasor_from_lifetime(
    frequency, component_lifetimes
)

plot = PhasorPlot(
    frequency=frequency, title='Graphical analysis of three components'
)
plot.hist2d(real, imag, cmap='Greys')
plot.components(
    component_real, component_imag, labels=['A', 'B', 'C'], **component_style
)
plot.show()

The results of the graphical component analysis are shown as line plots for each component pair:

counts = phasor_component_graphical(
    real,
    imag,
    component_real,
    component_imag,
    fractions=fractions,
    radius=radius,
)

fig, ax = pyplot.subplots()
ax.plot(fractions, counts[0], '-', label='A vs B')
ax.plot(fractions, counts[1], '-', label='A vs C')
ax.plot(fractions, counts[2], '-', label='B vs C')
ax.set_title('Graphical analysis of three components')
ax.set_xlabel('Fraction')
ax.set_ylabel('Count')
ax.legend()
pyplot.show()

The graphical method for resolving the contributions of three components (pairwise) to phasor coordinates is based on counting phasor coordinates within moving circular cursors along the line between the components, demonstrated in the following animation for component A vs B. To complete the analysis, the process is repeated for the other combinations of components, A vs C and B vs C:

fig, (ax, hist) = pyplot.subplots(nrows=2, ncols=1, figsize=(5.5, 8))

plot = PhasorPlot(
    frequency=frequency,
    ax=ax,
    title='Graphical analysis of component A vs B',
)
plot.hist2d(real, imag, cmap='Greys')
plot.components(
    component_real[:2],
    component_imag[:2],
    labels=['A', 'B'],
    **component_style,
)
plot.components(
    component_real[2], component_imag[2], labels=['C'], **component_style
)

hist.set_xlim(0, 1)
hist.set_xlabel('Fraction')
hist.set_ylabel('Count')

direction_real = component_real[0] - component_real[1]
direction_imag = component_imag[0] - component_imag[1]

plots = []
for i in range(fractions.size):
    cursor_real = component_real[1] + fractions[i] * direction_real
    cursor_imag = component_imag[1] + fractions[i] * direction_imag
    plot_lines = plot.plot(
        [cursor_real, component_real[2]],
        [cursor_imag, component_imag[2]],
        '-',
        linewidth=plot.dataunit_to_point * radius * 2 + 5,
        solid_capstyle='round',
        color='red',
        alpha=0.5,
    )
    hist_artists = pyplot.plot(
        fractions[: i + 1], counts[0][: i + 1], linestyle='-', color='tab:blue'
    )
    plots.append(plot_lines + hist_artists)

_ = matplotlib.animation.ArtistAnimation(fig, plots, interval=100, blit=True)
pyplot.tight_layout()
pyplot.show()

Once Loop Reflect