126 lines
5.0 KiB
Python
126 lines
5.0 KiB
Python
"""
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Geometric picture of the ROLL TPR@FPR objective.
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Upper panel: score PDFs for both classes; shaded area under the positive-class
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PDF to the left of τ equals the ROLL loss F̂₁(τ).
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Lower panel: CDFs showing how τ is read from F̂₀ and how the loss and TPR
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are read from F̂₁.
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Run from the thesis root:
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nix develop --command python3 content/method/figures/roll_principle.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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from scipy.stats import norm
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OUT_DIR = "content/method/figures"
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np.random.seed(3)
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# --- Parameters ------------------------------------------------------------
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mu0, sig0 = 0.0, 1.00 # negative class N(0, 1)
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mu1, sig1 = 1.5, 1.00 # positive class N(1.5, 1) — intentional overlap
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alpha = 0.20 # target FPR
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# τ = (1-α)-quantile of the negative class → F̂₀(τ) = 1-α → FPR = α
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tau = norm.ppf(1 - alpha, mu0, sig0)
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F0_tau = norm.cdf(tau, mu0, sig0) # = 1 - alpha
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F1_tau = norm.cdf(tau, mu1, sig1) # ROLL loss
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tpr = 1.0 - F1_tau
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# --- Sample points for positive class (rug) --------------------------------
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rug = np.random.normal(mu1, sig1, 35)
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# --- Score axis ------------------------------------------------------------
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x = np.linspace(-3.6, 5.0, 900)
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# --- Palette ---------------------------------------------------------------
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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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C_LOSS = "#7799BB"
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# ── Figure ─────────────────────────────────────────────────────────────────
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fig, (ax1, ax2) = plt.subplots(
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2, 1, figsize=(5.5, 4.8), sharex=True,
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gridspec_kw={"height_ratios": [1.25, 1.0]},
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)
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fig.subplots_adjust(hspace=0.07)
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# ══ Top: PDF ══════════════════════════════════════════════════════════════
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pdf0 = norm.pdf(x, mu0, sig0)
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pdf1 = norm.pdf(x, mu1, sig1)
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ax1.plot(x, pdf0, color=C_NEG, linewidth=1.8, label=r"Negative class ($y=0$)")
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ax1.plot(x, pdf1, color=C_POS, linewidth=1.8, label=r"Positive class ($y=1$)")
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ax1.axvline(tau, color=C_TAU, linestyle="--", linewidth=1.3,
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label=r"Threshold $\tau$")
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# Shade: area under positive PDF to the left of τ = loss = F̂₁(τ)
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x_left = x[x <= tau]
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ax1.fill_between(x_left, norm.pdf(x_left, mu1, sig1),
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color=C_LOSS, alpha=0.30, label=r"Loss $= \hat{F}_1(\tau)$")
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# Rug plot
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rug_y = np.full_like(rug, -0.006)
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ax1.scatter(rug, rug_y, color=C_POS, s=18, marker="|",
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zorder=3, alpha=0.65, clip_on=False)
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ax1.set_ylabel("PDF", fontsize=9)
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ax1.set_yticks([])
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ax1.set_ylim(bottom=-0.025)
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ax1.legend(fontsize=7.5, loc="upper right", framealpha=0.88)
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# ══ Bottom: CDF ═══════════════════════════════════════════════════════════
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cdf0 = norm.cdf(x, mu0, sig0)
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cdf1 = norm.cdf(x, mu1, sig1)
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ax2.plot(x, cdf0, color=C_NEG, linewidth=1.8, label=r"$\hat{F}_0$ (neg.)")
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ax2.plot(x, cdf1, color=C_POS, linewidth=1.8, label=r"$\hat{F}_1$ (pos.)")
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ax2.axvline(tau, color=C_TAU, linestyle="--", linewidth=1.3)
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# Horizontal reference lines
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ax2.axhline(F0_tau, color=C_NEG, linestyle="-.", linewidth=0.9, alpha=0.7)
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ax2.axhline(F1_tau, color=C_POS, linestyle="-.", linewidth=0.9, alpha=0.7)
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# Dots at the intercepts
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ax2.plot(tau, F0_tau, "o", color=C_NEG, markersize=5, zorder=4)
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ax2.plot(tau, F1_tau, "o", color=C_POS, markersize=5, zorder=4)
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# Right-side labels for the intercepts (placed where CDFs have flattened to ~1)
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x_label_right = x[-1] - 0.2
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ax2.text(x_label_right, F0_tau, rf"$1-\alpha$",
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fontsize=8, color=C_NEG, va="bottom", ha="right")
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ax2.text(x_label_right, F1_tau, r"$\hat{F}_1(\tau)$" + "\n(loss)",
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fontsize=7.5, color=C_POS, va="top", ha="right")
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# TPR annotation: double-headed arrow on the right + label
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x_bracket = tau + 1.9
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ax2.annotate("", xy=(x_bracket, 1.0), xytext=(x_bracket, F1_tau),
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arrowprops=dict(arrowstyle="<->", color="black",
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lw=0.8, mutation_scale=10))
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ax2.text(x_bracket + 0.12, (1.0 + F1_tau) / 2,
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f"TPR\n= {tpr*100:.0f}%",
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fontsize=7.5, va="center", ha="left", color="black")
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ax2.set_ylabel("CDF", fontsize=9)
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ax2.set_xlabel(r"Score $f_\theta(\mathbf{x})$", fontsize=9)
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ax2.set_yticks([0.0, 0.25, 0.50, 0.75, 1.0])
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ax2.set_yticklabels(["0", "0.25", "0.5", "0.75", "1"], fontsize=7.5)
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ax2.set_xlim(x[0] - 0.3, x[-1])
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ax2.legend(fontsize=7.5, loc="upper left", framealpha=0.88)
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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}/roll_principle.pdf", bbox_inches="tight")
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fig.savefig(f"{OUT_DIR}/roll_principle.png", bbox_inches="tight", dpi=180)
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print(f"Saved to {OUT_DIR}/roll_principle.{{pdf,png}}")
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print(f"tau={tau:.3f} F0(tau)={F0_tau:.3f} F1(tau)={F1_tau:.3f} TPR={tpr:.3f}")
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