# skew_normal_explorer_app.py # Run locally: # pip install -r requirements.txt # python skew_normal_explorer_app.py import numpy as np import gradio as gr import matplotlib.pyplot as plt from math import sqrt, pi from scipy.stats import skewnorm def theoretical_stats_skewnorm(alpha, mu, sigma): """ Compute theoretical moments for the Skew-Normal(alpha, loc=mu, scale=sigma). Returns mean, median, mode (numeric), variance, std dev, IQR, skewness, kurtosis (Pearson) & excess kurtosis. """ m, v, s, exk = skewnorm.stats(alpha, loc=mu, scale=sigma, moments="mvsk") m = float(m); v = float(v); s = float(s); exk = float(exk) std = float(np.sqrt(v)) # Quartiles q1 = float(skewnorm.ppf(0.25, alpha, loc=mu, scale=sigma)) q3 = float(skewnorm.ppf(0.75, alpha, loc=mu, scale=sigma)) iqr = q3 - q1 # Median median = float(skewnorm.ppf(0.5, alpha, loc=mu, scale=sigma)) # Mode via grid search xs = np.linspace(m - 6*std, m + 6*std, 4001) pdf_vals = skewnorm.pdf(xs, alpha, loc=mu, scale=sigma) mode = float(xs[int(np.argmax(pdf_vals))]) return { "mean": m, "median": median, "mode": mode, "variance": v, "std_dev": std, "IQR": iqr, "range": float("inf"), "skewness": s, "kurtosis": exk + 3.0, "excess_kurtosis": exk } def sample_stats(sample): n = len(sample) if n < 2: s_mean = float(sample[0]) if n == 1 else float("nan") return { "mean": s_mean, "median": s_mean, "mode": s_mean, "variance": 0.0, "std_dev": 0.0, "IQR": 0.0, "range": 0.0, "skewness": 0.0, "kurtosis": 3.0, "excess_kurtosis": 0.0 } s = np.asarray(sample, dtype=float) s_mean = float(np.mean(s)) s_median = float(np.median(s)) # Mode estimate via histogram counts, bin_edges = np.histogram(s, bins=min(50, max(5, int(np.sqrt(n))))) max_bin_idx = int(np.argmax(counts)) mode_est = float((bin_edges[max_bin_idx] + bin_edges[max_bin_idx+1]) / 2.0) s_var = float(np.var(s, ddof=1)) s_std = float(np.sqrt(s_var)) q1, q3 = np.percentile(s, [25, 75]) iqr = q3 - q1 s_range = float(np.max(s) - np.min(s)) m2 = np.mean((s - s_mean)**2) m3 = np.mean((s - s_mean)**3) m4 = np.mean((s - s_mean)**4) if m2 <= 0: skew, kurt = 0.0, 3.0 else: skew = m3 / (m2 ** 1.5) kurt = m4 / (m2 ** 2) ex_kurt = kurt - 3.0 return { "mean": s_mean, "median": s_median, "mode": mode_est, "variance": s_var, "std_dev": s_std, "IQR": iqr, "range": s_range, "skewness": skew, "kurtosis": kurt, "excess_kurtosis": ex_kurt } def format_stats_block(title, d): range_str = "∞" if d["range"] == float("inf") else f"{d['range']:.6g}" lines = [ f"**{title}**", f"- Mean: {d['mean']:.6g}", f"- Median: {d['median']:.6g}", f"- Mode: {d['mode']:.6g}", f"- Variance: {d['variance']:.6g}", f"- Std Dev: {d['std_dev']:.6g}", f"- IQR: {d['IQR']:.6g}", f"- Range: {range_str}", f"- Skewness: {d['skewness']:.6g}", f"- Kurtosis: {d['kurtosis']:.6g}", f"- Excess Kurtosis: {d['excess_kurtosis']:.6g}", ] return "\n".join(lines) def render(alpha, mu, sigma, n, seed, x_min, x_max, bins, show_hist, overlay_empirical_pdf): sigma = max(1e-6, sigma) if x_min >= x_max: theo_tmp = theoretical_stats_skewnorm(alpha, mu, sigma) m, std = theo_tmp["mean"], max(1e-9, theo_tmp["std_dev"]) x_min, x_max = m - 4*std, m + 4*std x = np.linspace(x_min, x_max, 800) y = skewnorm.pdf(x, alpha, loc=mu, scale=sigma) rng = np.random.default_rng(int(seed)) sample = skewnorm.rvs(alpha, loc=mu, scale=sigma, size=int(n), random_state=rng) theo = theoretical_stats_skewnorm(alpha, mu, sigma) samp = sample_stats(sample) fig, ax = plt.subplots(figsize=(8, 4.5), dpi=120) ax.plot(x, y, label="Theoretical PDF (Skew-Normal)") if show_hist: ax.hist(sample, bins=int(bins), density=True, alpha=0.5, label="Sample histogram") if overlay_empirical_pdf: bw = 1.06 * max(1e-8, samp["std_dev"]) * (len(sample) ** (-1/5)) bw = max(bw, 1e-6) diffs = (x.reshape(-1, 1) - sample.reshape(1, -1)) / bw kernel_vals = np.exp(-0.5 * diffs**2) / (sqrt(2*pi) * bw) kde = np.mean(kernel_vals, axis=1) ax.plot(x, kde, linestyle="--", label="Empirical density (KDE-like)") ax.set_title("Skew-Normal & Normal Explorer (α=0 gives Normal)") ax.set_xlabel("x") ax.set_ylabel("density") ax.legend(loc="best") ax.grid(True, linestyle="--") left = format_stats_block("Theoretical (Skew-Normal)", theo) right = format_stats_block("Sample (from sliders)", samp) stats_md = left + "\n\n" + right return fig, stats_md with gr.Blocks(title="Skew-Normal & Normal Explorer") as demo: gr.Markdown("# Skew-Normal & Normal Explorer") gr.Markdown( "Slide **α (skewness)** to skew left/right. **α=0 → Normal(μ, σ²)**. " "Adjust μ, σ, n, and window. Compare theoretical vs sample stats." ) with gr.Row(): with gr.Column(scale=1): alpha = gr.Slider(-15.0, 15.0, value=0.0, step=0.1, label="Skewness (α)") mu = gr.Slider(-10.0, 10.0, value=0.0, step=0.1, label="Location (μ)") sigma = gr.Slider(0.1, 10.0, value=1.0, step=0.1, label="Scale (σ)") n = gr.Slider(10, 200000, value=2000, step=10, label="Sample size (n)") seed = gr.Slider(0, 99999, value=42, step=1, label="Random seed") with gr.Accordion("Plot window & layers", open=False): x_min = gr.Number(value=-5.0, label="x min") x_max = gr.Number(value=5.0, label="x max") bins = gr.Slider(5, 200, value=40, step=1, label="Histogram bins") show_hist = gr.Checkbox(value=True, label="Show sample histogram") overlay_empirical_pdf = gr.Checkbox(value=False, label="Overlay empirical density (KDE-like)") with gr.Column(scale=2): plot = gr.Plot(label="Curve / Histogram") stats = gr.Markdown(label="Descriptive Statistics") inputs = [alpha, mu, sigma, n, seed, x_min, x_max, bins, show_hist, overlay_empirical_pdf] demo.load(render, inputs=inputs, outputs=[plot, stats]) for w in inputs: w.change(render, inputs=inputs, outputs=[plot, stats]) if __name__ == "__main__": demo.launch()