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DescriptionSample mean and variance of IID samples from a standard Cauchy distribution..png |
English: sample mean and variance of IID samples from a standard Cauchy distribution.
```python import numpy as np import matplotlib.pyplot as plt def running_sample_variance(data): """ Welford's online algorithm to calculate the variance. In each iteration, the mean and M2 (the sum of the squares of differences from the mean) are updated. For each column i, the difference delta between the current data point and the previous mean is calculated, and the mean and M2 are updated accordingly. variance = M2 / np.arange(1, m + 1). """ n, m = data.shape mean = np.zeros((n, m)) M2 = np.zeros((n, m)) for i in range(m): if i == 0: delta = data[:, i] - mean[:, i] mean[:, i] = mean[:, i] + delta / (i + 1) M2[:, i] = delta * (data[:, i] - mean[:, i]) else: delta = data[:, i] - mean[:, i - 1] mean[:, i] = mean[:, i - 1] + delta / (i + 1) M2[:, i] = M2[:, i - 1] + delta * (data[:, i] - mean[:, i]) variance_n = M2 / (np.arange(1, m + 1)) return variance_n def running_sample_average(data): cumsum = np.cumsum(data, axis=1) cum_n = np.arange(1, data.shape[1] + 1) return cumsum / cum_n
n = 10 # Number of rows m = 500000 # Number of columns
data = np.random.standard_cauchy(size=(n, m)) variances = running_sample_variance(data) mean = running_sample_average(data)
fig, (ax2, ax1) = plt.subplots(1, 2, figsize=(16, 5))
x = np.arange(1, variances.shape[1] + 1) ax1.semilogy(variances.T, linewidth=1) ax1.set_xlabel('Number of Samples') ax1.set_ylabel('Variance') ax1.set_title('Running Sample Variances')
ax2.semilogy(np.abs(mean.T), linewidth=1) ax2.set_xlabel('Number of Samples') ax2.set_ylabel('Average (absolute value)') ax2.set_title('Running Sample Averages')
plt.tight_layout() plt.show() ``` |
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Source | Own work |
Author | Cosmia Nebula |
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current | 21:17, 2 June 2023 | 1,589 × 490 (115 KB) | Cosmia Nebula | Uploaded while editing "Cauchy distribution" on en.wikipedia.org |
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