Improvements! And another figure.

content/method/figures/roll_principle.py

@@ -0,0 +1,125 @@
+"""
+Geometric picture of the ROLL TPR@FPR objective.
+
+Upper panel: score PDFs for both classes; shaded area under the positive-class
+PDF to the left of τ equals the ROLL loss F̂₁(τ).
+Lower panel: CDFs showing how τ is read from F̂₀ and how the loss and TPR
+are read from F̂₁.
+
+Run from the thesis root:
+    nix develop --command python3 content/method/figures/roll_principle.py
+"""
+
+import numpy as np
+import matplotlib
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt
+from scipy.stats import norm
+
+OUT_DIR = "content/method/figures"
+
+np.random.seed(3)
+
+# --- Parameters ------------------------------------------------------------
+mu0, sig0 = 0.0, 1.00    # negative class N(0, 1)
+mu1, sig1 = 1.5, 1.00    # positive class N(1.5, 1)  — intentional overlap
+alpha = 0.20             # target FPR
+
+# τ = (1-α)-quantile of the negative class  →  F̂₀(τ) = 1-α  →  FPR = α
+tau = norm.ppf(1 - alpha, mu0, sig0)
+F0_tau = norm.cdf(tau, mu0, sig0)   # = 1 - alpha
+F1_tau = norm.cdf(tau, mu1, sig1)   # ROLL loss
+tpr = 1.0 - F1_tau
+
+# --- Sample points for positive class (rug) --------------------------------
+rug = np.random.normal(mu1, sig1, 35)
+
+# --- Score axis ------------------------------------------------------------
+x = np.linspace(-3.6, 5.0, 900)
+
+# --- Palette ---------------------------------------------------------------
+C_NEG  = "#4477AA"
+C_POS  = "#CC6633"
+C_TAU  = "#333333"
+C_LOSS = "#7799BB"
+
+# ── Figure ─────────────────────────────────────────────────────────────────
+fig, (ax1, ax2) = plt.subplots(
+    2, 1, figsize=(5.5, 4.8), sharex=True,
+    gridspec_kw={"height_ratios": [1.25, 1.0]},
+)
+fig.subplots_adjust(hspace=0.07)
+
+# ══ Top: PDF ══════════════════════════════════════════════════════════════
+
+pdf0 = norm.pdf(x, mu0, sig0)
+pdf1 = norm.pdf(x, mu1, sig1)
+
+ax1.plot(x, pdf0, color=C_NEG, linewidth=1.8, label=r"Negative class ($y=0$)")
+ax1.plot(x, pdf1, color=C_POS, linewidth=1.8, label=r"Positive class ($y=1$)")
+ax1.axvline(tau, color=C_TAU, linestyle="--", linewidth=1.3,
+            label=r"Threshold $\tau$")
+
+# Shade: area under positive PDF to the left of τ = loss = F̂₁(τ)
+x_left = x[x <= tau]
+ax1.fill_between(x_left, norm.pdf(x_left, mu1, sig1),
+                 color=C_LOSS, alpha=0.30, label=r"Loss $= \hat{F}_1(\tau)$")
+
+# Rug plot
+rug_y = np.full_like(rug, -0.006)
+ax1.scatter(rug, rug_y, color=C_POS, s=18, marker="|",
+            zorder=3, alpha=0.65, clip_on=False)
+
+ax1.set_ylabel("PDF", fontsize=9)
+ax1.set_yticks([])
+ax1.set_ylim(bottom=-0.025)
+ax1.legend(fontsize=7.5, loc="upper right", framealpha=0.88)
+
+# ══ Bottom: CDF ═══════════════════════════════════════════════════════════
+
+cdf0 = norm.cdf(x, mu0, sig0)
+cdf1 = norm.cdf(x, mu1, sig1)
+
+ax2.plot(x, cdf0, color=C_NEG, linewidth=1.8, label=r"$\hat{F}_0$ (neg.)")
+ax2.plot(x, cdf1, color=C_POS, linewidth=1.8, label=r"$\hat{F}_1$ (pos.)")
+ax2.axvline(tau, color=C_TAU, linestyle="--", linewidth=1.3)
+
+# Horizontal reference lines
+ax2.axhline(F0_tau, color=C_NEG, linestyle="-.", linewidth=0.9, alpha=0.7)
+ax2.axhline(F1_tau, color=C_POS, linestyle="-.", linewidth=0.9, alpha=0.7)
+
+# Dots at the intercepts
+ax2.plot(tau, F0_tau, "o", color=C_NEG, markersize=5, zorder=4)
+ax2.plot(tau, F1_tau, "o", color=C_POS, markersize=5, zorder=4)
+
+# Left-side y-axis labels for the intercepts
+x_min = x[0]
+ax2.text(x_min - 0.15, F0_tau, rf"$1-\alpha$",
+         fontsize=8, color=C_NEG, va="center", ha="right")
+ax2.text(x_min - 0.15, F1_tau, r"$\hat{F}_1(\tau)$" + "\n(loss)",
+         fontsize=7.5, color=C_POS, va="center", ha="right")
+
+# TPR annotation: double-headed arrow on the right + label
+x_bracket = tau + 1.9
+ax2.annotate("", xy=(x_bracket, 1.0), xytext=(x_bracket, F1_tau),
+             arrowprops=dict(arrowstyle="<->", color="black",
+                             lw=0.8, mutation_scale=10))
+ax2.text(x_bracket + 0.12, (1.0 + F1_tau) / 2,
+         f"TPR\n= {tpr*100:.0f}%",
+         fontsize=7.5, va="center", ha="left", color="black")
+
+ax2.set_ylabel("CDF", fontsize=9)
+ax2.set_xlabel(r"Score $f_\theta(\mathbf{x})$", fontsize=9)
+ax2.set_yticks([0.0, 0.25, 0.50, 0.75, 1.0])
+ax2.set_yticklabels(["0", "0.25", "0.5", "0.75", "1"], fontsize=7.5)
+ax2.set_xlim(x[0] - 0.3, x[-1])
+ax2.legend(fontsize=7.5, loc="upper left", framealpha=0.88)
+
+for ax in (ax1, ax2):
+    ax.tick_params(labelsize=8)
+    ax.spines[["top", "right"]].set_visible(False)
+
+fig.savefig(f"{OUT_DIR}/roll_principle.pdf", bbox_inches="tight")
+fig.savefig(f"{OUT_DIR}/roll_principle.png", bbox_inches="tight", dpi=180)
+print(f"Saved to {OUT_DIR}/roll_principle.{{pdf,png}}")
+print(f"tau={tau:.3f}  F0(tau)={F0_tau:.3f}  F1(tau)={F1_tau:.3f}  TPR={tpr:.3f}")

content/method/method.tex

@@ -251,6 +251,23 @@ of the two populations, turning a TPR@FPR objective into a FPR@TPR one. Because
 two objectives are therefore interchangeable at the implementation level, all derivations
 and implementation details below are given for \eqref{eq:roll-tpr-at-fpr} only;
 \eqref{eq:roll-fpr-at-tpr} follows by the same transformation applied to the inputs.
+\Cref{fig:roll-principle} provides a geometric view of \eqref{eq:roll-tpr-at-fpr}.
+
+\begin{figure}[H]
+  \centering
+  \includegraphics[width=0.85\textwidth]{content/method/figures/roll_principle.png}
+  \caption{Geometric picture of the ROLL TPR@FPR objective. \textit{Upper}: score PDFs
+    for the negative (blue, $y=0$) and positive (orange, $y=1$) classes. The operating
+    threshold $\tau$ (dashed) is placed at the $(1-\alpha)$-quantile of the negative
+    class, so that exactly $\alpha$ fraction of negative scores exceed it ($\operatorname{FPR}=\alpha$).
+    The shaded area under the positive-class PDF to the left of $\tau$ is
+    $\hat{F}_1(\tau)$ --- the ROLL loss we minimize. Tick marks along the bottom represent
+    observed positive-class scores. \textit{Lower}: corresponding CDFs $\hat{F}_0$ (blue)
+    and $\hat{F}_1$ (orange). Reading horizontally at height $1-\alpha$ on $\hat{F}_0$
+    gives $\tau$; reading $\hat{F}_1(\tau)$ off $\hat{F}_1$ at the same $\tau$ gives the
+    loss; the distance from $\hat{F}_1(\tau)$ to $1$ is the resulting TPR.}
+  \label{fig:roll-principle}
+\end{figure}
 
 We now derive the gradient of \eqref{eq:roll-tpr-at-fpr} with respect to $f_\theta(\mathbf{x}_i)$.