josh
6f977c11cf
在既有 K/U 消元关系中分离完整信息损失、角度协方差及合成指标。 明确单星冻结可分模型的正交条件，并用消参考 Doppler 后的耦合解释多星近重合。 保留有限距离相对率的角导数和完整协方差变换，不扩大模型或实验范围。 复用现有确定性脚本核对协方差更新、SS 正交性与非零信息损失反例。 自检及 SyncTeX 多遍编译通过，无断引用或新增溢出；实际图值以代码侧回灌为准。
318 lines
14 KiB
Python
318 lines
14 KiB
Python
"""Deterministic formula checks for Appendices F/G; run: python check_finite_range_bearing.py.
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Synthetic sphere and known trajectories only: no constellation or MC validation.
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"""
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import numpy as np
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RADIUS = 6_371_000.0
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WAVELENGTH = 10.0
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SPIN = np.array([[0., -7.292115e-5, 0.], [7.292115e-5, 0., 0.], [0., 0., 0.]])
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TIMES = np.array([0., .13, .29, .51, .83])
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DESIGN = np.column_stack((np.ones(len(TIMES)), TIMES))
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ARRAY = np.array([[0., 0., 0.], [1., 0., 0.], [0., 1., 0.], [1., 1., 0.]])
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def direction(x):
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a, b = x
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ca, sa, cb, sb = np.cos(a), np.sin(a), np.cos(b), np.sin(b)
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return np.array([cb*ca, cb*sa, sb]), np.array([
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[-cb*sa, -sb*ca], [cb*ca, -sb*sa], [0., cb]])
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GROUND, _ = direction(np.array([.35, .42]))
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GROUND *= RADIUS
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SATELLITES = np.array([GROUND*1.10 + [90_000., -80_000., 40_000.],
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GROUND*1.11 + [-210_000., 230_000., -60_000.]])
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VELOCITIES = np.array([[1100., 6800., -1400.], [-1800., 6200., 2700.]])
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ACCELERATIONS = np.array([[-4., -3., -1.], [-3., -2., -2.]])
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def bearing(p, satellite):
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e = p - satellite
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e /= np.linalg.norm(e)
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return np.array([np.arctan2(e[1], e[0]), np.arcsin(e[2])])
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def intersection(x):
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e, E = direction(x)
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s = SATELLITES[0]
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disc = (s@e)**2 - (s@s - RADIUS**2)
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assert disc > 0, "Ray is tangent or misses the surface"
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d = -s@e - np.sqrt(disc)
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assert d > 0
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p = s + d*e
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n = p / RADIUS
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assert n@e < 0, "Expected near visible branch"
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P = d*(np.eye(3) - np.outer(e, n)/(n@e))@E
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return p, P
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def geometry(x, t, ell):
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p0, P0 = intersection(x)
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angle = 7.292115e-5*t
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c, s = np.cos(angle), np.sin(angle)
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B = np.array([[c, -s, 0.], [s, c, 0.], [0., 0., 1.]])
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p, P = B@p0, B@P0
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v = VELOCITIES[ell] + t*ACCELERATIONS[ell] - SPIN@p
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a = ACCELERATIONS[ell] - SPIN@SPIN@p
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Vx, Ax = -SPIN@P, -SPIN@SPIN@P
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rvec = SATELLITES[ell] + t*VELOCITIES[ell] + .5*t*t*ACCELERATIONS[ell] - p
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r = np.linalg.norm(rvec)
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w = rvec/r
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S = np.eye(3) - np.outer(w, w)
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W = -S@P/r
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wd = S@v/r
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rx = -w@P
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WD = np.column_stack([
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-(np.outer(W[:, j], w) + np.outer(w, W[:, j]))@v/r
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+ S@Vx[:, j]/r - wd*rx[j]/r for j in range(2)
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])
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b = -v@w/WAVELENGTH
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bd = -(a@w + v@wd)/WAVELENGTH
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bx = -(w@Vx + v@W)/WAVELENGTH
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bdx = -(w@Ax + a@W + wd@Vx + v@WD)/WAVELENGTH
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return w, W, wd, WD, b, bx, bd, bdx
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def mixed_motion(x, t, ell):
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"""Synthetic non-spin rotation: deliberately not an astronomical model."""
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p0, P0 = intersection(x)
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a, k = 7.292115e-5*t, 3e-5
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z = np.array([[np.cos(a), -np.sin(a), 0.],
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[np.sin(a), np.cos(a), 0.], [0., 0., 1.]])
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a = k*t
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q = np.array([[1., 0., 0.], [0., np.cos(a), -np.sin(a)],
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[0., np.sin(a), np.cos(a)]])
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generator = np.array([[0., 0., 0.], [0., 0., -k], [0., k, 0.]])
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B = z@q
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Bd = SPIN@B + B@generator
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C, Cd = SPIN@B, SPIN@Bd
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p, pd, vu = B@p0, Bd@p0, C@p0
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s = SATELLITES[ell] + t*VELOCITIES[ell] + .5*t*t*ACCELERATIONS[ell]
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vs = VELOCITIES[ell] + t*ACCELERATIONS[ell]
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r = np.linalg.norm(s-p)
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w = (s-p)/r
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S = np.eye(3)-np.outer(w, w)
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wd = S@(vs-pd)/r
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proxy = S@(vs-vu)/r
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W = -S@B@P0/r
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b = -(vs-vu)@w/WAVELENGTH
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bg = -(vs-pd)@w/WAVELENGTH
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bd = -((ACCELERATIONS[ell]-Cd@p0)@w + (vs-vu)@wd)/WAVELENGTH
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bx = -(-w@C@P0+(vs-vu)@W)/WAVELENGTH
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return p, pd, vu, w, wd, proxy, b, bg, bd, bx
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def coefficients(x, ell):
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data = [geometry(x, t, ell) for t in TIMES]
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ref = [geometry(x, t, 0) for t in TIMES]
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c = np.linalg.lstsq(DESIGN, [a[4]-b[4] for a, b in zip(data, ref)], rcond=None)[0]
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Cx = np.linalg.lstsq(DESIGN, [a[5]-b[5] for a, b in zip(data, ref)], rcond=None)[0]
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return c, Cx
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def mean_and_jacobian(x, ref_coeff, ell):
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c, Cx = coefficients(x, ell)
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f, rate = ref_coeff + c
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w, W = geometry(x, 0., ell)[:2]
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steering = np.exp(-2j*np.pi*ARRAY@w/WAVELENGTH)
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Ax = (-2j*np.pi/WAVELENGTH)*steering[:, None]*(ARRAY@W)
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q = np.exp(2j*np.pi*(f*TIMES + .5*rate*TIMES**2))
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qx = q[:, None]*2j*np.pi*(TIMES[:, None]*Cx[0] + .5*TIMES[:, None]**2*Cx[1])
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Jx = np.column_stack([np.kron(q, Ax[:, j])+np.kron(qx[:, j], steering) for j in range(2)])
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Jf = np.kron(2j*np.pi*TIMES*q, steering)
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Jr = np.kron(1j*np.pi*TIMES**2*q, steering)
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return np.kron(q, steering), np.column_stack((Jx, Jf, Jr))
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def finite_difference(f, x, step=2e-7):
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return np.stack([(f(x+step*e)-f(x-step*e))/(2*step) for e in np.eye(len(x))], axis=-1)
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def check(name, actual, expected, tolerance=2e-7):
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error = np.linalg.norm(actual-expected)/max(1., np.linalg.norm(expected))
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assert error < tolerance, (name, error, tolerance)
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print(f"{name}: relative error {error:.3g}")
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def main():
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x = bearing(GROUND, SATELLITES[0])
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p, P = intersection(x)
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check("ray intersection", p, GROUND)
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check("intersection Jacobian", P, finite_difference(lambda x: intersection(x)[0], x))
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latlon = np.array([.35, .42])
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_, Pll = direction(latlon)
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Pll *= RADIUS
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e = (p-SATELLITES[0])/np.linalg.norm(p-SATELLITES[0])
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K = np.array([[-e[1]/(e[0]**2+e[1]**2), e[0]/(e[0]**2+e[1]**2), 0.],
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[0., 0., 1/np.sqrt(1-e[2]**2)]])
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H = K@(np.eye(3)-np.outer(e, e))@Pll/np.linalg.norm(p-SATELLITES[0])
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check("bearing/latlon tangent identity", P@H, Pll)
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for ell in range(2):
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w, W, wd, WD, b, bx, bd, bdx = geometry(x, .41, ell)
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for name, index, jac in [("LOS", 0, W), ("LOS time derivative", 2, WD),
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("Doppler", 4, bx), ("rate", 6, bdx)]:
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check(name, jac, finite_difference(lambda x: geometry(x, .41, ell)[index], x))
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step = 1e-3
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check("rate time derivative", bd,
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(geometry(x, .41+step, ell)[4]-geometry(x, .41-step, ell)[4])/(2*step))
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c, Cx = coefficients(x, 1)
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check("OLS coefficient Jacobian", Cx, finite_difference(lambda x: coefficients(x, 1)[0], x))
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z = np.r_[x, 20., .8]
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means, jac = zip(*(mean_and_jacobian(x, z[2:], ell) for ell in range(2)))
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J = np.vstack(jac)
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check("CP full mean Jacobian", J,
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finite_difference(lambda z: np.concatenate([mean_and_jacobian(z[:2], z[2:], ell)[0]
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for ell in range(2)]), z), 2e-6)
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T = np.zeros((4, 4))
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T[:2, :2] = H
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T[2:, 2:] = np.eye(2)
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Jll = finite_difference(lambda z: np.concatenate([
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mean_and_jacobian(bearing(RADIUS*direction(z[:2])[0], SATELLITES[0]), z[2:], ell)[0]
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for ell in range(2)]), np.r_[latlon, z[2:]], step=1e-8)
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check("direct versus chain Jacobian", J@T, Jll, 2e-6)
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A = np.vstack((J.real, J.imag))*np.sqrt(2)
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nuisance = np.zeros((len(A), 4))
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for ell in range(2):
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gain = np.concatenate([means[i] if i == ell else np.zeros_like(means[i]) for i in range(2)])
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nuisance[:, 2*ell] = np.r_[gain.real, gain.imag]*np.sqrt(2)
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nuisance[:, 2*ell+1] = np.r_[-gain.imag, gain.real]*np.sqrt(2)
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projected = (np.eye(len(A))-nuisance@np.linalg.pinv(nuisance))@A
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info = projected.T@projected
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assert np.linalg.matrix_rank(projected) == 4
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cov = np.linalg.inv(info)
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check("full covariance reparameterization", T@np.linalg.inv(T.T@info@T)@T.T, cov)
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# Appendix F: structured CP has real frame amplitudes and one phase per star.
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for selected in [[0], [0, 1]]:
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Jcp = np.vstack([jac[ell] for ell in selected])
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N = np.zeros((len(Jcp), len(selected)*(len(TIMES)+1)), dtype=complex)
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block_size = len(TIMES)*len(ARRAY)
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for star, ell in enumerate(selected):
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rows = slice(star*block_size, (star+1)*block_size)
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N[rows, star*(len(TIMES)+1)] = 1j*means[ell]
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for frame in range(len(TIMES)):
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block = slice(star*block_size+frame*len(ARRAY),
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star*block_size+(frame+1)*len(ARRAY))
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N[block, star*(len(TIMES)+1)+frame+1] = means[ell][frame*len(ARRAY):(frame+1)*len(ARRAY)]
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real_N = np.vstack((N.real, N.imag))
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scores = np.vstack((Jcp.real, Jcp.imag))*np.sqrt(2)
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residual_scores = scores-real_N@np.linalg.lstsq(real_N, scores, rcond=None)[0]
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I = residual_scores.T@residual_scores
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K, coupling, rate_info = I[:3, :3], I[:3, 3], I[3, 3]
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CK = np.linalg.inv(K)
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CU = np.linalg.inv(K-np.outer(coupling, coupling)/rate_info)
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v = CK@coupling
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h = rate_info-coupling@v
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assert h > 0, "K/U covariance identities require identifiable rate"
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check("CP K/U full covariance update", CU-CK, np.outer(v, v)/h, 1e-9)
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q = coupling[:2]-K[:2, 2]*coupling[2]/K[2, 2]
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G = K[:2, :2]-np.outer(K[:2, 2], K[:2, 2])/K[2, 2]
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check("partial angle-rate coupling", v[:2], np.linalg.solve(G, q), 1e-9)
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metric = np.diag([np.cos(x[1])**2, 1.])
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check("CP K/U angular metric update", np.trace(metric@(CU[:2, :2]-CK[:2, :2])),
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v[:2]@metric@v[:2]/h, 1e-9)
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if len(selected) == 1:
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check("SS spatial-temporal orthogonality", I[:2, 2:], np.zeros((2, 2)), 1e-10)
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check("SS exact angular subblock equality", CU[:2, :2], CK[:2, :2], 1e-10)
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assert CU[2, 2] > CK[2, 2], "First-sample time origin couples frequency and rate"
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# Nonzero EFIM loss can affect only Doppler after the covariance update.
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K = np.array([[3., .3, .5], [.3, 2., -.4], [.5, -.4, 4.]])
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CK = np.linalg.inv(K)
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for v in [np.array([0., 0., .7]), np.array([.2, -.3, .7])]:
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coupling = K@v
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rate_info = coupling@v+2.
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CU = np.linalg.inv(K-np.outer(coupling, coupling)/rate_info)
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check("synthetic rate-only covariance update", CU-CK, np.outer(v, v)/2., 1e-12)
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assert np.linalg.norm(coupling[:2]) > 0
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if np.linalg.norm(v[:2]) == 0:
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check("nonzero EFIM loss with equal angle bounds", CU[:2, :2], CK[:2, :2], 1e-12)
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assert CU[2, 2] > CK[2, 2]
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else:
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assert np.trace(metric@(CU[:2, :2]-CK[:2, :2])) > 0
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delta = .19
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shifted = TIMES-delta
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for ell in range(2):
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f, rate = z[2:]+coefficients(x, ell)[0]
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phase0 = 2*np.pi*(f*TIMES+.5*rate*TIMES**2)
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phase1 = 2*np.pi*(f*delta+.5*rate*delta**2
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+(f+delta*rate)*shifted+.5*rate*shifted**2)
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check("epoch phase identity", np.exp(1j*phase0), np.exp(1j*phase1))
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def switch(z):
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p, _ = intersection(z[:2])
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return np.r_[bearing(p, SATELLITES[1]), z[2:]+coefficients(z[:2], 1)[0]]
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Tref = np.eye(4)
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Tref[:2, :2] = finite_difference(lambda x: bearing(intersection(x)[0], SATELLITES[1]), x)
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Tref[2:, :2] = Cx
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check("reference-change full Jacobian", Tref, finite_difference(switch, z), 2e-6)
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for ell in range(2):
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p, pd, vu, w, wd, proxy, b, bg, bd, bx = mixed_motion(x, .41, ell)
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step = 1e-3
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check("mixed position velocity", pd,
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(mixed_motion(x, .41+step, ell)[0]-mixed_motion(x, .41-step, ell)[0])/(2*step),
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2e-6)
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check("mixed true LOS time derivative", wd,
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(mixed_motion(x, .41+step, ell)[3]-mixed_motion(x, .41-step, ell)[3])/(2*step))
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check("mixed node Doppler Jacobian", bx,
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finite_difference(lambda x: mixed_motion(x, .41, ell)[6], x))
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check("mixed Doppler time derivative", bd,
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(mixed_motion(x, .41+step, ell)[6]-mixed_motion(x, .41-step, ell)[6])/(2*step))
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check("mixed Doppler discrepancy", b-bg, (vu-pd)@w/WAVELENGTH)
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check("mixed proxy LOS discrepancy", proxy-wd,
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-(np.eye(3)-np.outer(w, w))@(vu-pd)/np.linalg.norm(
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SATELLITES[ell]+.41*VELOCITIES[ell]+.5*.41**2*ACCELERATIONS[ell]-p))
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assert np.linalg.norm(vu-pd) > 1, "Fixture must expose the mixed protocol"
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e = -w
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check("orientation steering identity",
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np.exp(-2j*np.pi*ARRAY@w/WAVELENGTH),
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np.exp(2j*np.pi*ARRAY@e/WAVELENGTH))
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check("orientation Doppler identity", b, (VELOCITIES[ell]+.41*ACCELERATIONS[ell]-vu)@e/WAVELENGTH)
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mixed_coeff = lambda x: np.linalg.lstsq(DESIGN, [
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mixed_motion(x, t, 1)[6]-mixed_motion(x, t, 0)[6] for t in TIMES], rcond=None)[0]
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mixed_jac = np.linalg.lstsq(DESIGN, [
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mixed_motion(x, t, 1)[9]-mixed_motion(x, t, 0)[9] for t in TIMES], rcond=None)[0]
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check("mixed fixed-design OLS Jacobian", mixed_jac, finite_difference(mixed_coeff, x))
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# Appendix G: independent block phases remove constants, not intra-block slopes.
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for nodes, starts, expected_rank in [([0.], [0., 3.], 0),
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([0., 2.], [0.], 1),
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([0., 2.], [0., 3.], 2),
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([0., 2.], [0., 3e-4], 2)]:
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blocks, weights, centers = [], [], []
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n = np.array(nodes)
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for ell in range(2):
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f, rate = z[2:]+coefficients(x, ell)[0]
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for start in starts:
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w = geometry(x, start, ell)[0]
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a = np.exp(-2j*np.pi*ARRAY@w/WAVELENGTH)
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mu = np.kron(np.exp(2j*np.pi*((f+rate*start)*n+.5*rate*n*n)), a)
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J = np.column_stack((1j*np.repeat(n, len(a))*mu,
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1j*np.repeat(start*n+.5*n*n, len(a))*mu))
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projected = J-mu[:, None]*(mu.conj()@J/(mu.conj()@mu))[None, :]
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expected = np.column_stack((
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1j*np.repeat(n-n.mean(), len(a))*mu,
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1j*np.repeat(start*(n-n.mean())+.5*(n*n-(n*n).mean()), len(a))*mu))
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check("IP centered temporal scores", projected, expected)
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blocks.append(projected)
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weights.append(2*np.linalg.norm(projected[:, 0])**2)
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centers.append(start+(n[-1]/2))
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J = np.vstack(blocks)
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real_J = np.vstack((J.real, J.imag))*np.sqrt(2)
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assert np.linalg.matrix_rank(real_J, tol=1e-10) == expected_rank
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if len(n) == 2:
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weights, centers = np.array(weights), np.array(centers)
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I = real_J.T@real_J
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vectors = np.column_stack((np.ones(len(centers)), centers))
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check("two-sample temporal information", I, vectors.T@(weights[:, None]*vectors))
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residual = real_J[:, 0]-real_J[:, 1]*(I[1, 0]/I[1, 1])
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check("real rate Schur complement", residual@residual,
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I[0, 0]-I[0, 1]**2/I[1, 1])
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if expected_rank == 1:
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check("single-frame exact frequency/rate cancellation", J@np.array([1., -1.]),
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np.zeros(len(J)))
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elif starts[-1] < 1e-3:
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assert 0 < residual@residual/I[0, 0] < 1e-7
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print("PASS: synthetic local formula and temporal-rank checks; code-side geometry and CP/IP acceptance still required.")
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if __name__ == "__main__":
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main()
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