113 lines
2.5 KiB
Markdown
113 lines
2.5 KiB
Markdown
---
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title: Decrease in Sigma
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tags:
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- destiny/uncertain
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- type/encyclopedia-entry
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---
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# Decrease in Sigma
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%%
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relevant to [[statistics]]:
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graphs intended to visually demonstrate
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a decrease in [[uncertainty]]
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for a random variable represented by a probability distribution.
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%%
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## Normal PDF
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```tikz
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\usepackage{pgfplots}
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\pgfplotsset{compat=1.16}
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\begin{document}
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\begin{tikzpicture}
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\begin{axis}[
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width=13cm,
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height=7cm,
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axis lines=middle,
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xlabel={$x$},
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ylabel={$\varphi(x;\mu,\sigma)$},
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xmin=-6, xmax=6,
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ymin=0, ymax=0.85,
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samples=400,
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domain=-6:6,
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legend style={draw=none, fill=none, at={(0.98,0.98)}, anchor=north east},
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legend cell align=left,
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ytick=\empty,
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]
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% Normal PDF: (1/(sigma*sqrt(2*pi))) * exp(-(x-mu)^2/(2*sigma^2))
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\addplot[thick]
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{ (1/(1.8*sqrt(2*pi))) * exp(-((x-0.8)^2)/(2*1.8^2)) };
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\addlegendentry{$\mu=0,\ \sigma=1.8$}
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\addplot[thick, dashed]
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{ (1/(0.8*sqrt(2*pi))) * exp(-((x-0.8)^2)/(2*0.8^2)) };
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\addlegendentry{$\mu=0,\ \sigma=0.8$}
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\end{axis}
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\end{tikzpicture}
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\end{document}
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```
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## Lognormal PDF
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![[lognormal-pdf.gif]]
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%%
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```python
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import numpy as np
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import matplotlib.pyplot as plt
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from scipy.stats import lognorm
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import imageio.v2 as imageio
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import os
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mean_target = 10.0
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sigmas = np.linspace(1.0, 0.1, 25)
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x = np.linspace(0.001, 40, 2000)
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# Precompute global y-limit
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pdf_max = 0.0
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for sigma in sigmas:
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mu = np.log(mean_target) - 0.5 * sigma**2
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pdf = lognorm.pdf(x, s=sigma, scale=np.exp(mu))
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pdf_max = max(pdf_max, pdf.max())
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frames = []
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tmp_dir = "frames"
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os.makedirs(tmp_dir, exist_ok=True)
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for i, sigma in enumerate(sigmas):
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mu = np.log(mean_target) - 0.5 * sigma**2
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pdf = lognorm.pdf(x, s=sigma, scale=np.exp(mu))
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plt.figure(figsize=(6, 4))
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plt.plot(x, pdf)
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plt.axvline(mean_target, linestyle="--", linewidth=1)
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plt.text(
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mean_target, pdf_max * 0.95,
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"mean",
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rotation=90,
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verticalalignment="top",
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horizontalalignment="right"
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)
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plt.title(f"Lognormal PDF\nmean = {mean_target}, sigma = {sigma:f}")
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plt.xlabel("x")
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plt.ylabel("density")
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plt.ylim(0, pdf_max * 1.05)
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plt.tight_layout()
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frame_path = f"{tmp_dir}/frame_{i:02d}.png"
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plt.savefig(frame_path, dpi=120)
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plt.close()
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frames.append(imageio.imread(frame_path))
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os.makedirs("out", exist_ok=True)
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gif_path = "out/lognormal-pdf.gif"
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imageio.mimsave(gif_path, frames, duration=0.15)
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```
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%% |