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October 15, 2019 04:42
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Rotation preserves probability
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| import numpy as np | |
| from numpy.linalg import norm | |
| from scipy.special import erf | |
| def get_rotation_matrix(x): | |
| """ | |
| Get rotation matrix for the space that aligns vector x with y = [1, 0, 0, 0, 0, ..., 0] | |
| See: https://math.stackexchange.com/questions/598750/finding-the-rotation-matrix-in-n-dimensions | |
| """ | |
| u = x / norm(x) | |
| y = np.zeros_like(u) | |
| y[0] = 1. | |
| v = y - np.dot(u, y) * u | |
| v /= norm(v) | |
| cost = np.dot(x, y) / (norm(x) * norm(y)) | |
| sint = np.sqrt(1. - cost**2.) | |
| outer_v = np.outer(v, v) | |
| outer_u = np.outer(u, u) | |
| Rt = np.array([[cost, -sint], [sint, cost]]) | |
| uv = np.vstack([u, v]) | |
| R = np.eye(x.shape[0]) - outer_v - outer_u + uv.T.dot(Rt).dot(uv) | |
| assert np.sum(np.abs(R.dot(u) - y)) < 1e-12 | |
| assert np.sum(np.abs(R.dot(R.T) - np.eye(x.shape[0]))) < 1e-12 | |
| return R | |
| def experiment(mean, sigma, num_points): | |
| ## Run a simple experiment to determine F1 Score empirically | |
| pts = np.random.multivariate_normal(np.zeros_like(mean), np.eye(mean.shape[0])*(sigma**2), num_points) | |
| distance_to_origin = norm(pts, axis=1) | |
| distance_to_mean = norm(pts - mean, axis=1) | |
| FP = np.sum(distance_to_mean < distance_to_origin) / num_points | |
| F1 = 1. - FP | |
| ## Show that the constructed rotation of data points preserves F1 Score | |
| R = get_rotation_matrix(mean) | |
| rotated_pts = pts.dot(R) | |
| # now we only need to look at the first dimesion to determine classification outcome | |
| rotated_FP = np.sum(rotated_pts[:, 0] > norm(mean) / 2.) / num_points | |
| rotated_F1 = 1. - rotated_FP | |
| return F1, rotated_F1 | |
| def exact_soln(mean, sigma): | |
| return 0.5 * (1. + erf(norm(mean) / (2. * np.sqrt(2) * sigma))) | |
| def main(): | |
| mean_vec = np.random.randn(100) + 0.5 | |
| sigma = 5. | |
| F1, rotated_F1 = experiment(mean_vec, sigma, 5000000) | |
| exact_F1 = exact_soln(mean_vec, sigma) | |
| print("F1 score in original space: %.8f" % F1) | |
| print("F1 score in rotated space : %.8f" % rotated_F1) | |
| print("Exact Analytical Solution : %.8f" % exact_F1) | |
| if __name__ == '__main__': | |
| main() |
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