109 lines
4.2 KiB
Python
109 lines
4.2 KiB
Python
"""
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Illustrates the gradient locality property of KDE-ROLL.
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Upper panel: score distributions for both classes with operating threshold.
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Lower panel: gradient magnitude per sample, showing bell-shaped decay from threshold.
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Run from the thesis root:
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nix develop --command python3 content/method/figures/threshold_weighting.py
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"""
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import numpy as np
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import matplotlib
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matplotlib.use("Agg")
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import matplotlib.pyplot as plt
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# --- Add these configuration rules right here ---
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plt.rcParams.update({
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"text.usetex": False, # Rely on Matplotlib's math parser instead of the system compiler
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"mathtext.fontset": "cm", # Emulate standard LaTeX Computer Modern styles flawlessly
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"pdf.fonttype": 42 # Ensure fonts are embedded as TrueType vector curves instead of Type 3 paths
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})
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OUT_DIR = "content/method/figures"
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np.random.seed(7)
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# --- Data ------------------------------------------------------------------
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n = 120
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alpha = 0.25 # target FPR
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neg_scores = np.sort(np.random.normal(-0.6, 0.75, n))
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pos_scores = np.sort(np.random.normal(1.1, 0.80, n))
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# Threshold: alpha-quantile of the negative-class scores
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tau = np.percentile(neg_scores, 100 * (1 - alpha))
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# --- Bandwidth (Silverman's rule per class) --------------------------------
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h_neg = 1.06 * neg_scores.std() * n ** (-1 / 5)
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h_pos = 1.06 * pos_scores.std() * n ** (-1 / 5)
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def sigma_prime(x, h):
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"""Logistic kernel PDF: derivative of the sigmoid-smoothed CDF."""
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e = np.exp(-np.abs(x) / h)
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return e / (h * (1 + e) ** 2)
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# --- KDE-ROLL gradients (eq:kde-grad-combined) ----------------------------
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# y=1: (1/n) * sigma_prime(score - tau)
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grad_pos = (1.0 / n) * sigma_prime(pos_scores - tau, h_pos)
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# y=0: -(scale) * sigma_prime(tau - score_i) / sum_j[sigma_prime(tau - score_j)]
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# where scale = (1/n) * sum_j[sigma_prime(pos_j - tau)]
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neg_kernel = sigma_prime(tau - neg_scores, h_neg)
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scale = (1.0 / n) * np.sum(sigma_prime(pos_scores - tau, h_pos))
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grad_neg_signed = -scale * neg_kernel / neg_kernel.sum()
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grad_neg_abs = np.abs(grad_neg_signed)
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# --- Smooth reference bell curves -----------------------------------------
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x_ref = np.linspace(neg_scores.min() - 0.3, pos_scores.max() + 0.3, 400)
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bell_pos = sigma_prime(x_ref - tau, h_pos)
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bell_pos *= (grad_pos.max() / bell_pos.max())
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bell_neg = sigma_prime(tau - x_ref, h_neg)
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bell_neg *= (grad_neg_abs.max() / bell_neg.max())
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# --- Plot -----------------------------------------------------------------
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C_NEG = "#4477AA"
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C_POS = "#CC6633"
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C_TAU = "#333333"
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fig, (ax1, ax2) = plt.subplots(
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2, 1, figsize=(5.5, 4.2), sharex=True,
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gridspec_kw={"height_ratios": [1.3, 1.0]}
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)
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fig.subplots_adjust(hspace=0.06)
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# — Top: score distributions —
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ax1.hist(neg_scores, bins=20, density=True, alpha=0.45, color=C_NEG,
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label=r"Negative class ($y=0$)")
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ax1.hist(pos_scores, bins=20, density=True, alpha=0.45, color=C_POS,
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label=r"Positive class ($y=1$)")
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ax1.axvline(tau, color=C_TAU, linestyle="--", linewidth=1.4,
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label=rf"Threshold $\tau$ (FPR $=\alpha={alpha}$)")
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ax1.set_ylabel("Density", fontsize=9)
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ax1.set_yticks([])
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ax1.legend(fontsize=7.5, loc="upper right", framealpha=0.85)
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# — Bottom: gradient magnitudes —
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ax2.scatter(neg_scores, grad_neg_abs, color=C_NEG, s=12, zorder=3,
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alpha=0.75, label=r"$|\partial\mathcal{L}/\partial f_\theta(\mathbf{x})|$, $y=0$")
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ax2.scatter(pos_scores, grad_pos, color=C_POS, s=12, zorder=3,
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alpha=0.75, label=r"$\partial\mathcal{L}/\partial f_\theta(\mathbf{x})$, $y=1$")
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ax2.plot(x_ref, bell_neg, color=C_NEG, linewidth=1.2, alpha=0.5)
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ax2.plot(x_ref, bell_pos, color=C_POS, linewidth=1.2, alpha=0.5)
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ax2.axvline(tau, color=C_TAU, linestyle="--", linewidth=1.4)
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ax2.set_ylabel("Gradient magnitude", fontsize=9)
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ax2.set_xlabel(r"Score $f_\theta(\mathbf{x})$", fontsize=9)
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ax2.set_yticks([])
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ax2.legend(fontsize=7.5, loc="upper right", framealpha=0.85)
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for ax in (ax1, ax2):
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ax.tick_params(labelsize=8)
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ax.spines[["top", "right"]].set_visible(False)
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fig.savefig(f"{OUT_DIR}/threshold_weighting.pdf", bbox_inches="tight")
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fig.savefig(f"{OUT_DIR}/threshold_weighting.png", bbox_inches="tight", dpi=180)
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print(f"Saved to {OUT_DIR}/threshold_weighting.{{pdf,png}}")
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print(f"Sum neg: {np.sum(grad_neg_signed)}, pos: {np.sum(grad_pos)}")
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