Thesis - wip

content/method/figures/threshold_weighting_gaussian.py

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+"""
+Illustrates the gradient locality property of Gaussian-ROLL.
+
+Upper panel: score distributions for both classes with fitted Gaussian PDFs
+             and operating threshold.
+Lower panel: gradient magnitude per sample using the Gaussian gradient formula
+             (eq:gauss-grad-combined).
+
+Run from the thesis root:
+    nix develop --command python3 content/method/figures/threshold_weighting_gaussian.py
+"""
+
+import numpy as np
+from scipy import stats
+import matplotlib
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt
+
+plt.rcParams.update({
+    "text.usetex": False,
+    "mathtext.fontset": "cm",
+    "pdf.fonttype": 42
+})
+
+OUT_DIR = "content/method/figures"
+
+np.random.seed(7)
+
+# --- Data ------------------------------------------------------------------
+n = 120
+alpha = 0.25  # target FPR
+
+neg_scores = np.sort(np.random.normal(-0.6, 0.75, n))
+pos_scores = np.sort(np.random.normal(1.1, 0.80, n))
+
+n0 = len(neg_scores)
+n1 = len(pos_scores)
+
+# --- Gaussian MLE parameters -----------------------------------------------
+mu0, sigma0 = neg_scores.mean(), neg_scores.std()
+mu1, sigma1 = pos_scores.mean(), pos_scores.std()
+
+# Threshold: (1-alpha)-quantile of negative class so that FPR = alpha
+# (FPR = P(score > tau | y=0) = 1 - F_0(tau), so F_0(tau) = 1-alpha)
+tau = stats.norm.ppf(1 - alpha, loc=mu0, scale=sigma0)
+
+# Shared scalar: pdf of positive class evaluated at the threshold
+common = (1.0 / (sigma1 * np.sqrt(2 * np.pi))) * np.exp(-0.5 * ((tau - mu1) / sigma1) ** 2)
+
+# --- Gaussian ROLL gradients (eq:gauss-grad-combined) ---------------------
+# y=1: -(1/n1) * common * (1 + (tau-mu1)*(s-mu1)/sigma1^2)
+grad_pos_signed = -(1.0 / n1) * common * (
+    1 + (tau - mu1) * (pos_scores - mu1) / sigma1 ** 2
+)
+grad_pos_abs = np.abs(grad_pos_signed)
+
+# y=0: +(1/n0) * common * (1 + (tau-mu0)*(s-mu0)/sigma0^2)
+grad_neg_signed = (1.0 / n0) * common * (
+    1 + (tau - mu0) * (neg_scores - mu0) / sigma0 ** 2
+)
+grad_neg_abs = np.abs(grad_neg_signed)
+
+# Gradient balance check (should be ~0)
+print(f"Gradient balance: pos sum = {grad_pos_signed.sum():.6f}, "
+      f"neg sum = {grad_neg_signed.sum():.6f}, "
+      f"total = {(grad_pos_signed.sum() + grad_neg_signed.sum()):.2e}")
+
+# --- Smooth reference curves (continuous gradient functions) ---------------
+x_ref = np.linspace(neg_scores.min() - 0.3, pos_scores.max() + 0.3, 400)
+smooth_pos_abs = np.abs(-(1.0 / n1) * common * (
+    1 + (tau - mu1) * (x_ref - mu1) / sigma1 ** 2
+))
+smooth_neg_abs = np.abs((1.0 / n0) * common * (
+    1 + (tau - mu0) * (x_ref - mu0) / sigma0 ** 2
+))
+
+# --- Plot -----------------------------------------------------------------
+C_NEG = "#4477AA"
+C_POS = "#CC6633"
+C_TAU = "#333333"
+
+fig, (ax1, ax2) = plt.subplots(
+    2, 1, figsize=(5.5, 4.2), sharex=True,
+    gridspec_kw={"height_ratios": [1.3, 1.0]}
+)
+fig.subplots_adjust(hspace=0.06)
+
+# — Top: score distributions with fitted Gaussian PDFs —
+ax1.hist(neg_scores, bins=20, density=True, alpha=0.35, color=C_NEG,
+         label=r"Negative class ($y=0$)")
+ax1.hist(pos_scores, bins=20, density=True, alpha=0.35, color=C_POS,
+         label=r"Positive class ($y=1$)")
+ax1.plot(x_ref, stats.norm.pdf(x_ref, mu0, sigma0),
+         color=C_NEG, linewidth=1.8, label=r"Fitted $\mathcal{N}(\mu_0,\sigma_0^2)$")
+ax1.plot(x_ref, stats.norm.pdf(x_ref, mu1, sigma1),
+         color=C_POS, linewidth=1.8, label=r"Fitted $\mathcal{N}(\mu_1,\sigma_1^2)$")
+ax1.axvline(tau, color=C_TAU, linestyle="--", linewidth=1.4,
+            label=rf"Threshold $\tau$  (FPR $=\alpha={alpha}$)")
+ax1.set_ylabel("Density", fontsize=9)
+ax1.set_yticks([])
+ax1.legend(fontsize=7, loc="upper right", framealpha=0.85)
+
+# — Bottom: gradient magnitudes —
+ax2.scatter(neg_scores, grad_neg_abs, color=C_NEG, s=12, zorder=3,
+            alpha=0.75, label=r"$|\partial\mathcal{L}/\partial f_\theta(\mathbf{x})|$, $y=0$")
+ax2.scatter(pos_scores, grad_pos_abs, color=C_POS, s=12, zorder=3,
+            alpha=0.75, label=r"$|\partial\mathcal{L}/\partial f_\theta(\mathbf{x})|$, $y=1$")
+ax2.plot(x_ref, smooth_neg_abs, color=C_NEG, linewidth=1.2, alpha=0.55)
+ax2.plot(x_ref, smooth_pos_abs, color=C_POS, linewidth=1.2, alpha=0.55)
+ax2.axvline(tau, color=C_TAU, linestyle="--", linewidth=1.4)
+ax2.set_ylabel("Gradient magnitude", fontsize=9)
+ax2.set_xlabel(r"Score $f_\theta(\mathbf{x})$", fontsize=9)
+ax2.set_yticks([])
+ax2.legend(fontsize=7.5, loc="upper right", framealpha=0.85)
+
+for ax in (ax1, ax2):
+    ax.tick_params(labelsize=8)
+    ax.spines[["top", "right"]].set_visible(False)
+
+out_base = f"{OUT_DIR}/threshold_weighting_gaussian"
+fig.savefig(f"{out_base}.pdf", bbox_inches="tight")
+fig.savefig(f"{out_base}.png", bbox_inches="tight", dpi=180)
+print(f"Saved to {out_base}.{{pdf,png}}")