7.6. Polarimeter field serialization

Import Python libraries Hide code cell content

from __future__ import annotations

import json
import logging
import math
import os

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from ampform.sympy import perform_cached_doit
from ampform_dpd.io import perform_cached_lambdify
from IPython.display import Markdown
from scipy.interpolate import RegularGridInterpolator, griddata
from tqdm.auto import tqdm

from polarimetry import formulate_polarimetry
from polarimetry.data import (
    create_data_transformer,
    generate_meshgrid_sample,
    generate_phasespace_sample,
)
from polarimetry.io import (
    export_polarimetry_field,
    import_polarimetry_field,
    mute_jax_warnings,
)
from polarimetry.lhcb import load_model_builder, load_model_parameters
from polarimetry.lhcb.particle import load_particles
from polarimetry.plot import use_mpl_latex_fonts

mute_jax_warnings()

model_choice = "Default amplitude model"
model_file = "../../data/model-definitions.yaml"
particles = load_particles("../../data/particle-definitions.yaml")
amplitude_builder = load_model_builder(model_file, particles, model_choice)
imported_parameter_values = load_model_parameters(
    model_file, amplitude_builder.decay, model_choice, particles
)
reference_subsystem = 1
model = amplitude_builder.formulate(reference_subsystem)
model.parameter_defaults.update(imported_parameter_values)

NO_LOG = "EXECUTE_NB" in os.environ
if NO_LOG:
    logging.disable(logging.CRITICAL)

Formulate expressions and lambdify Hide code cell source

polarimetry_exprs = formulate_polarimetry(amplitude_builder, reference_subsystem)
unfolded_exprs = [
    perform_cached_doit(expr.doit().xreplace(model.amplitudes))
    for expr in tqdm(
        [model.full_expression, *polarimetry_exprs], disable=NO_LOG, leave=False
    )
]
actual_funcs = [
    perform_cached_lambdify(expr.xreplace(model.parameter_defaults), backend="jax")
    for expr in tqdm(unfolded_exprs, leave=False, desc="Lambdifying", disable=NO_LOG)
]

7.6.1. File size checks

Export do different file formats Hide code cell source

alpha_x_func = actual_funcs[1]
alpha_x = alpha_x_func(grid_sample).real
df = pd.DataFrame(alpha_x, index=X[0], columns=Y[:, 0])
os.makedirs("export", exist_ok=True)
df.to_json("export/alpha-x-pandas.json")
df.to_json("export/alpha-x-pandas-json.zip", compression={"method": "zip"})
df.to_csv("export/alpha-x-pandas.csv")

df_dict = df.to_dict()
filtered_df_dict = {
    x: {y: v for y, v in row.items() if not math.isnan(v)} for x, row in df_dict.items()
}
with open("export/alpha-x-python.json", "w") as f:
    json.dump(filtered_df_dict, f)

json_dict = dict(
    x=X[0].tolist(),
    y=Y[:, 0].tolist(),
    z=alpha_x.tolist(),
)
with open("export/alpha-x-arrays.json", "w") as f:
    json.dump(json_dict, f, separators=(",", ":"))

File sizes for 100x100 grid:

File type	Size
export/alpha-x-arrays.json	141 kB
export/alpha-x-pandas.json	311 kB
export/alpha-x-python.json	260 kB
export/alpha-x-pandas-json.zip	51 kB
export/alpha-x-pandas.csv	128 kB

7.6.2. Export polarimetry grids

Decided to use the alpha-x-arrays.json format. It can be exported with export_polarimetry_field().

os.makedirs("export", exist_ok=True)
filename = "export/polarimetry-model-0.json"
export_polarimetry_field(
    sigma1=X[0],
    sigma2=Y[:, 0],
    intensity=actual_funcs[0](grid_sample).real,
    alpha_x=actual_funcs[1](grid_sample).real,
    alpha_y=actual_funcs[2](grid_sample).real,
    alpha_z=actual_funcs[3](grid_sample).real,
    filename=filename,
    metadata={"model description": model_choice},
)

Polarimetry grid can be downloaded here: export/polarimetry-model-0.json (540 kB).

7.6.3. Import and interpolate

The arrays in the exported JSON files can be used to create a RegularGridInterpolator for the intensity and for each components of \(\vec\alpha\).

import_polarimetry_field() returns JAX arrays, which are read-only. RegularGridInterpolator requires modifiable arrays, so we convert them to NumPy.

Also note that the values array needs to be transposed!

field_data = import_polarimetry_field("export/polarimetry-model-0.json")
imported_arrays = (
    field_data["intensity"],
    field_data["alpha_x"],
    field_data["alpha_y"],
    field_data["alpha_z"],
)
interpolated_funcs = [
    RegularGridInterpolator(
        points=(
            field_data["m^2_Kpi"],
            field_data["m^2_pK"],
        ),
        values=np.nan_to_num(z).transpose(),
        method="linear",
    )
    for z in imported_arrays
]

This is a function that can compute an interpolated value of each of these observables for a random point on the Dalitz plane.

interpolated_funcs[1]([0.8, 3.6])

array([0.18379986])

Note that RegularGridInterpolator is already in vectorized form, so there is no need to vectorize it.

n_points = 100_000
mini_sample = generate_phasespace_sample(model.decay, n_points, seed=0)
mini_sample = transformer(mini_sample)
x = mini_sample["sigma1"]
y = mini_sample["sigma2"]
z_interpolated = [func((x, y)) for func in tqdm(interpolated_funcs, disable=NO_LOG)]
z_interpolated[0]

array([2165.82154945, 5481.04128781, 6254.96174147, ..., 1369.40657535,
       4456.44114915, 7197.97782088], shape=(100000,))

Show code cell source Hide code cell source

use_mpl_latex_fonts()
plt.rc("font", size=18)
fig, axes = plt.subplots(
    dpi=200,
    figsize=(15, 11.5),
    gridspec_kw={"width_ratios": [1, 1, 1, 1.2]},
    ncols=4,
    nrows=3,
    sharex=True,
    sharey=True,
)
plt.subplots_adjust(hspace=0.1, wspace=0.03)
fig.suptitle("Comparison interpolated and actual values", y=0.94)

points = np.transpose([x, y])
for i in tqdm(range(4), disable=NO_LOG, leave=False):
    if i == 0:
        title = "$I$"
        cmap = plt.cm.viridis
        clim = None
    else:
        title = Rf"$\alpha_{'xyz'[i - 1]}$"
        cmap = plt.cm.coolwarm
        clim = (-1, +1)
    axes[0, i].set_title(title, y=1.03)

    z_actual = actual_funcs[i](mini_sample)
    z_diff = 100 * ((z_interpolated[i] - z_actual) / z_actual).real
    Z_interpolated = griddata(points, z_interpolated[i], (X, Y))
    Z_diff = griddata(points, z_diff, (X, Y))

    mesh = (
        axes[0, i].pcolormesh(X, Y, actual_funcs[i](grid_sample).real, cmap=cmap),
        axes[1, i].pcolormesh(X, Y, Z_interpolated, cmap=cmap),
        axes[2, i].pcolormesh(X, Y, Z_diff, clim=(-100, +100), cmap=plt.cm.coolwarm),
    )
    if i != 0:
        mesh[0].set_clim(-1, +1)
        mesh[1].set_clim(-1, +1)
    if i == 3:
        c_bar = [fig.colorbar(mesh[j], ax=axes[j, i], pad=0.015) for j in range(3)]
        c_bar[0].ax.set_ylabel("Original distribution", labelpad=3)
        c_bar[1].ax.set_ylabel("Interpolated distribution", labelpad=3)
        c_bar[2].ax.set_ylabel("Difference", labelpad=-20)
        for c in c_bar[:-1]:
            c.ax.set_yticks([-1, 0, +1])
            c.ax.set_yticklabels(["$-1$", "$0$", "$+1$"])
        c_bar[-1].ax.set_yticks([-100, 0, +100])
        c_bar[-1].ax.set_yticklabels([R"$-100\%$", R"$0\%$", R"$+100\%$"])
        axes[0, i].text(
            x=0.96,
            y=0.83,
            s=f"grid size:\n{resolution}x{resolution}",
            fontsize=16,
            horizontalalignment="right",
            transform=axes[0, i].transAxes,
        )
        axes[1, i].text(
            x=0.96,
            y=0.83,
            s=f"phsp size:\n{n_points:,}",
            fontsize=16,
            horizontalalignment="right",
            transform=axes[1, i].transAxes,
        )
plt.show()
plt.close(fig)

Note

The interpolated values over this phase space sample have been visualized by interpolating again over a meshgrid with scipy.interpolate.griddata.

Tip

Determination of polarization shows how this interpolation method can be used to determine the polarization \(\vec{P}\) from a given intensity distribution.