Plotting Results

Learning Objectives

By the end of this section you will understand:

  • How to plot two-time correlation heatmaps

  • How to plot fitted vs experimental C2

  • How to visualize parameter distributions from CMC

  • How to create publication-quality figures

—

Two-Time Correlation Heatmaps

The most important diagnostic visualization is the \(C_2(t_1, t_2)\) heatmap for each azimuthal angle:

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors as mcolors
from homodyne.data import load_xpcs_data

data = load_xpcs_data("config.yaml")
c2 = data['c2_exp']       # (n_phi, n_t1, n_t2)
t1 = data['t1']
t2 = data['t2']
phi = data['phi_angles_list']

n_phi = len(phi)
ncols = min(n_phi, 4)
nrows = (n_phi + ncols - 1) // ncols
fig, axes = plt.subplots(nrows, ncols, figsize=(4 * ncols, 4 * nrows),
                          squeeze=False)

for i_phi, phi_val in enumerate(phi):
    row, col = divmod(i_phi, ncols)
    ax = axes[row][col]

    # Plot C2 with log-scale time axes
    im = ax.pcolormesh(
        t1, t2, c2[i_phi].T,
        cmap='hot', vmin=0.95, vmax=1.5,
        shading='auto'
    )
    ax.set_xscale('log')
    ax.set_yscale('log')
    ax.set_xlabel("t1 (s)")
    ax.set_ylabel("t2 (s)")
    ax.set_title(f"φ = {phi_val:.0f}°")
    plt.colorbar(im, ax=ax, label="C2")

# Hide unused axes
for idx in range(n_phi, nrows * ncols):
    row, col = divmod(idx, ncols)
    axes[row][col].set_visible(False)

plt.suptitle("Two-Time Correlation Matrix", fontsize=14)
plt.tight_layout()
plt.savefig("c2_heatmaps.pdf", dpi=150, bbox_inches='tight')
plt.show()

—

Fitted vs Experimental Cuts

To inspect fit quality, compare one-dimensional cuts through \(C_2\):

import numpy as np
import matplotlib.pyplot as plt

# Extract the anti-diagonal (fixed lag time index = lag)
def extract_diagonal_cut(c2, lag=1):
    """Extract C2 values at fixed lag index (t2 - t1 = lag steps)."""
    n = min(c2.shape[0], c2.shape[1]) - lag
    return np.array([c2[k, k + lag] for k in range(n)])

c2_exp = data['c2_exp']   # (n_phi, n_t1, n_t2)
phi = data['phi_angles_list']
t1 = data['t1']

fig, ax = plt.subplots(figsize=(8, 5))
lags_to_plot = [1, 5, 20, 50]
colors = plt.cm.viridis(np.linspace(0, 1, len(lags_to_plot)))

for lag, color in zip(lags_to_plot, colors):
    for i_phi in range(len(phi)):
        cut = extract_diagonal_cut(c2_exp[i_phi], lag=lag)
        ax.plot(
            t1[:len(cut)], cut,
            color=color, alpha=0.3, linewidth=0.5
        )
ax.set_xscale('log')
ax.set_xlabel("t1 (s)")
ax.set_ylabel("C2")
ax.set_title("C2 diagonal cuts (all angles overlaid)")
plt.tight_layout()
plt.show()

—

NLSQ Fit Quality Plot

Visualize the NLSQ result alongside experimental data:

import numpy as np
import matplotlib.pyplot as plt

# After fitting:
result = fit_nlsq_jax(data, config)

# Extract relevant data for one angle
i_phi = 0  # First angle
c2_exp_slice = data['c2_exp'][i_phi]   # (n_t1, n_t2)
t1 = data['t1']
t2 = data['t2']

# Compute lag-time dependence (anti-diagonal)
lag_times = []
c2_exp_lag = []
max_lag = min(50, c2_exp_slice.shape[0] - 1)

for lag in range(1, max_lag):
    n = c2_exp_slice.shape[0] - lag
    c2_at_lag = np.array([c2_exp_slice[k, k+lag] for k in range(n)])
    lag_times.append(t1[lag] - t1[0])
    c2_exp_lag.append(np.mean(c2_at_lag))

fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(lag_times, c2_exp_lag, 'ko', markersize=3, label='Experiment')
# Model curve would be added here after computing with fitted parameters
ax.set_xscale('log')
ax.set_xlabel("Lag time (s)")
ax.set_ylabel("C2")
ax.set_title(f"Fit quality: chi^2_nu = {result.reduced_chi_squared:.3f}")
ax.legend()
plt.tight_layout()
plt.show()

—

CMC Posterior Plots

Use ArviZ for posterior visualization:

import arviz as az

idata = cmc_result.inference_data

# 1. Trace plots (time series of sampled values)
az.plot_trace(idata, var_names=["D0", "alpha"])
plt.suptitle("MCMC Trace Plots", y=1.02)
plt.tight_layout()
plt.savefig("trace_plots.pdf", bbox_inches='tight')

# 2. Posterior distributions
az.plot_posterior(
    idata,
    var_names=["D0", "alpha", "D_offset"],
    hdi_prob=0.95,
)
plt.suptitle("Posterior Distributions", y=1.02)
plt.tight_layout()
plt.savefig("posterior_distributions.pdf", bbox_inches='tight')

# 3. Pair plot (parameter correlations)
az.plot_pair(
    idata,
    var_names=["D0", "alpha", "gamma_dot_0"],
    divergences=True,    # Mark divergent transitions
    figsize=(8, 8),
)
plt.suptitle("Parameter Pair Plot", y=1.02)
plt.tight_layout()
plt.savefig("pair_plot.pdf", bbox_inches='tight')

—

Parameter Evolution Across Samples

Plot how a parameter (e.g., D₀) changes across a series of experiments:

import numpy as np
import matplotlib.pyplot as plt

# Collected results from batch processing
labels = ["Sample A", "Sample B", "Sample C", "Sample D"]
D0_vals = np.array([1245.3, 1389.2, 1102.8, 1560.4])
D0_errs = np.array([45.2, 67.8, 38.1, 89.5])

fig, ax = plt.subplots(figsize=(7, 4))
x = np.arange(len(labels))
ax.errorbar(x, D0_vals, yerr=D0_errs, fmt='o-', capsize=5,
            color='steelblue', linewidth=1.5, markersize=6)
ax.axhline(np.mean(D0_vals), color='gray', linestyle='--', label='Mean')
ax.fill_between(
    [-0.5, len(labels) - 0.5],
    [np.mean(D0_vals) - np.std(D0_vals)] * 2,
    [np.mean(D0_vals) + np.std(D0_vals)] * 2,
    alpha=0.2, color='gray', label='±1 std'
)
ax.set_xticks(x)
ax.set_xticklabels(labels)
ax.set_ylabel("D₀ (Ų/s)")
ax.set_title("Diffusion Coefficient Across Samples")
ax.legend()
plt.tight_layout()
plt.savefig("D0_comparison.pdf", bbox_inches='tight')

—

Publication-Quality Figures

For publication figures, set matplotlib style and use tight layouts:

import matplotlib as mpl
import matplotlib.pyplot as plt

# Use a publication-friendly style
plt.rcParams.update({
    'font.family': 'serif',
    'font.size': 10,
    'axes.labelsize': 11,
    'axes.titlesize': 11,
    'xtick.labelsize': 9,
    'ytick.labelsize': 9,
    'legend.fontsize': 9,
    'figure.dpi': 300,
    'axes.spines.top': False,
    'axes.spines.right': False,
})

# Single column width: 3.5 inches; double: 7 inches (typical journal)
fig, ax = plt.subplots(figsize=(3.5, 3.0))

# ... your plot ...

plt.tight_layout(pad=0.4)
plt.savefig("figure_1.pdf", bbox_inches='tight', dpi=300)

—

See Also