vault backup: 2026-01-11 12:50:45

decrease-in-sigma.md

@@ -11,6 +11,8 @@ tags:
 ---
 # Decrease in Sigma
 
+## Normal PDF
+
 ```tikz
 \usepackage{pgfplots}
 \pgfplotsset{compat=1.16}
@@ -47,53 +49,62 @@ tags:
 \end{document}
 ```
 
-^pdf
+## Lognormal PDF
 
-```tikz
-\usepackage{pgfplots}
-\pgfplotsset{compat=1.16}
+![[lognormal-pdf.gif]]
 
-\pgfmathdeclarefunction{erfapprox}{1}{%
-  \pgfmathparse{%
-    % save sign and work with |x|
-    ( (#1<0) ? -1 : 1 )
-    * ( 1 - (1 + 0.278393*abs(#1) + 0.230389*abs(#1)^2 + 0.000972*abs(#1)^3 + 0.078108*abs(#1)^4)^(-4) )
-  }%
-}
+%%
+```python
+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.stats import lognorm
+import imageio.v2 as imageio
+import os
 
-% Normal CDF using erf approximation:
-% F(x;mu,sigma) = 0.5*(1 + erf((x-mu)/(sigma*sqrt(2))))
-\pgfmathdeclarefunction{normcdf}{3}{%
-  \pgfmathparse{ 0.5*(1 + erfapprox((#1-#2)/(#3*sqrt(2)))) }%
-}
+mean_target = 10.0
+sigmas = np.linspace(1.0, 0.1, 25)
+x = np.linspace(0.001, 40, 2000)
 
-\begin{document}
-\begin{tikzpicture}
-    \begin{axis}[
-      width=13cm,
-      height=7cm,
-      axis lines=middle,
-      xlabel={$x$},
-      ylabel={$F(x;\mu,\sigma)$},
-      xmin=-6, xmax=6,
-      ymin=0, ymax=1.05,
-      samples=400,
-      domain=-6:6,
-      legend style={draw=none, fill=none, at={(0.02,0.98)}, anchor=north west},
-      legend cell align=left,
-      ytick={0,0.5,1},
-    ]
-    
-    % Normal CDF: 0.5*(1 + erf((x-mu)/(sigma*sqrt(2))))
-    \addplot[thick]
-      { normcdf(x,0,1.8) };
-    \addlegendentry{$\mu=0,\ \sigma=1.8$}
-    
-    \addplot[thick, dashed]
-      { normcdf(x,0.8) };
-    \addlegendentry{$\mu=0,\ \sigma=0.8$}
-    
-    \end{axis}
-\end{tikzpicture}
-\end{document}
+# Precompute global y-limit
+pdf_max = 0.0
+for sigma in sigmas:
+    mu = np.log(mean_target) - 0.5 * sigma**2
+    pdf = lognorm.pdf(x, s=sigma, scale=np.exp(mu))
+    pdf_max = max(pdf_max, pdf.max())
+
+frames = []
+tmp_dir = "frames"
+os.makedirs(tmp_dir, exist_ok=True)
+
+for i, sigma in enumerate(sigmas):
+    mu = np.log(mean_target) - 0.5 * sigma**2
+    pdf = lognorm.pdf(x, s=sigma, scale=np.exp(mu))
+
+    plt.figure(figsize=(6, 4))
+    plt.plot(x, pdf)
+    plt.axvline(mean_target, linestyle="--", linewidth=1)
+    plt.text(
+        mean_target, pdf_max * 0.95,
+        "mean",
+        rotation=90,
+        verticalalignment="top",
+        horizontalalignment="right"
+    )
+
+    plt.title(f"Lognormal PDF\nmean = {mean_target}, sigma = {sigma:f}")
+    plt.xlabel("x")
+    plt.ylabel("density")
+    plt.ylim(0, pdf_max * 1.05)
+    plt.tight_layout()
+
+    frame_path = f"{tmp_dir}/frame_{i:02d}.png"
+    plt.savefig(frame_path, dpi=120)
+    plt.close()
+
+    frames.append(imageio.imread(frame_path))
+
+os.makedirs("out", exist_ok=True)
+gif_path = "out/lognormal-pdf.gif"
+imageio.mimsave(gif_path, frames, duration=0.15)
 ```
+%%