Donsker_theorem_for_normal_distributions.gif
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Description Donsker theorem for normal distributions.gif |
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```python
import numpy as np import matplotlib.pyplot as plt import scipy import tempfile import os import imageio
def plot_bridge(samples=None, n=1000, dist=scipy.stats.norm, dist_name="normal", low=-3, high=+3): fig, axes = plt.subplot_mosaic("ABB", figsize=(20, 6)) ax1 = axes["A"] ax2 = axes["B"] if samples is None: samples = dist.rvs(size=n) else: n = len(samples) x = sorted(samples) y = dist.cdf(x) ref_x = np.linspace(low,high, 1000) ref_y = dist.cdf(ref_x) # calculate empirical cdf ecdf = np.arange(n) / len(samples) # plot cdf of standard normal distribution and empirical cdf of samples ax1.plot(ref_x, ref_y, label='Standard CDF') ax1.plot(x, ecdf, label='Empirical CDF') ax1.set_title() ax1.set_xlim(low,high) ax1.legend() ax1.set_ylim(0,1) ax2.plot([0.0] + dist.cdf(x) .tolist() + [1.0], [0.0] + (np.sqrt(n) * (ecdf - dist.cdf(x))).tolist() + [0.0]) ax2.set_title('centered, scaled, and re-timed') ax2.set_ylim(-0.9, 0.9) fig.suptitle(f"{dist_name}, with n = {n}") return fig def interpolate_counts(counts, frames_per_step): interpolated_counts = [counts[0]] for i in range(1,len(counts)): interval = (counts[i] - counts[i-1]) // i interpolated_counts += list(range(counts[i-1], counts[i], interval)) return interpolated_counts + [counts[-1]] with tempfile.TemporaryDirectory() as temp_dir: dist = scipy.stats.norm dist_name = "normal" low, high = -3.5, +3.5 n_steps = 16 frames_per_step = 10 sample_counts = interpolate_counts([2**n for n in range(n_steps)], frames_per_step) n_frames = len(sample_counts)-1 samples = dist.rvs(size=sample_counts[0]).tolist() for i in range(n_frames): samples += dist.rvs(size=sample_counts[i+1]-sample_counts[i]).tolist() fig = plot_bridge(samples, dist=dist, dist_name=dist_name, low=low,high=high) filename = os.path.join(temp_dir, f"plot_{i:03d}.png") fig.savefig(filename) plt.close(fig) # Compile images into GIF fps = 12 images = [] for i in range(n_frames): filename = os.path.join(temp_dir, f"plot_{i:03d}.png") images.append(imageio.imread(filename)) imageio.mimsave(f"{dist_name} Donsker theorem.gif", images, duration=1/fps)``` |
Date | |
Source | Own work |
Author | Cosmia Nebula |
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