716 lines
38 KiB
Python
716 lines
38 KiB
Python
import numpy as np
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import pywt
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from scipy.interpolate import griddata
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from scipy.stats import pearsonr, spearmanr
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import matplotlib.pyplot as plt
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# ---------------- 1. parameters ----------------
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rng = np.random.default_rng(0)
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# grid
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NX = NY = 64; N = NX*NY
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# ground truths
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BETA = 2.5 # potential-field truth: power spectrum ~ k^-BETA (before attenuation)
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Z_SRC = 1.5 # source depth (cells): upward-continuation attenuation exp(-2*pi*k*Z_SRC)
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N_MODES = 12 # Fourier-sparse truth: number of modes (low-frequency biased)
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# sampling patterns
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STRIDE = 3 # sparse (regular-grid) stride -> delta ~ 0.118
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NBR, NBA, RMAX, BLK = 45, 36, 90.0, 4 # BLK=4 ~ paper's patch size A/N_phi (2.9 px, non-integer)
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SA_ITERS = 4000
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# reconstruction solvers
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WAVELET, LEVELS, MODE = 'sym8', 2, 'periodization' # 2 = pywt.dwt_max_level(64, 'sym8')
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SYM_SPINS = 8 # cycle-spin shifts for the symlet reconstruction (our choice; the
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# paper never specifies cycle spinning or any solver parameters)
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POCS_ITERS, POCS_LMIN = 800, 1e-4
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# validation
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LOO_M = 60 # leave-one-out calibration points
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ENS_K = 64 # ensemble validation realizations (our construct, not the paper's)
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N_TRIALS = 25 # Monte-Carlo trials per truth class
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RUN_SANITY_CHECK = False # plt4 served its purpose; flip to True to re-run the check
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# ---------------- 2. ground truths (two tiers) ----------------
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# spectral grid shared by the truth generator and the Gaussian-process covariance
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KRAD = np.hypot(np.fft.fftfreq(NX)[:, None], np.fft.fftfreq(NY)[None, :])
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def power_spectrum(beta, z_src):
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# power spectrum k^-beta * exp(-4*pi*k*z): squared amplitude of the attenuated generator
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S = np.where(KRAD > 0, KRAD, 1.0)**(-beta)*np.exp(-4*np.pi*KRAD*z_src)
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S[0, 0] = 0.0
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return S
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def powerlaw_field(beta, z_src=Z_SRC):
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# random-phase power law with source-depth attenuation: without the exp(-2*pi*k*z) factor the
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# field is rough at pixel scale, which real (upward-continued) potential-field data never is
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amp = np.sqrt(power_spectrum(beta, z_src))
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f = np.real(np.fft.ifft2(amp*np.exp(2j*np.pi*rng.random((NX, NY)))))
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return (f - f.mean())/f.std()
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def fourier_sparse_field(n_modes):
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# visually structured sparse signal: amplitude ~ |k|^-1/2 so low frequencies dominate the look,
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# with half the modes beyond the sparse grid's Nyquist 1/(2*STRIDE) so that grid aliases
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k_alias = 1/(2*STRIDE)
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C = np.zeros((NX, NY), complex); picked = set()
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m = 0
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while m < n_modes:
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lo, hi = ((k_alias, 0.30) if m < n_modes//2 else (0.03, k_alias))
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kmag = np.exp(rng.uniform(np.log(lo), np.log(hi)))
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ang = rng.uniform(0, 2*np.pi)
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i = int(round(kmag*np.cos(ang)*NX)) % NX
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j = int(round(kmag*np.sin(ang)*NY)) % NY
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if (i, j) in picked or (i, j) == (0, 0):
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continue
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picked.add((i, j))
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C[i, j] = kmag**-0.5*np.exp(2j*np.pi*rng.random())
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m += 1
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f = np.fft.ifft2(C).real
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return (f - f.mean())/f.std()
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def white_noise_field():
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# incompressible control: no method can beat predicting zero at unsampled cells,
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# whose rms is sqrt(1 - DELTA) ~ 0.94
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f = rng.standard_normal((NX, NY))
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return (f - f.mean())/f.std()
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# ---------------- 3. sampling patterns and ISA properties (Eq. 4 ingredients) ----------------
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mask_reg = np.zeros((NX, NY), bool); mask_reg[::STRIDE, ::STRIDE] = True
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NS = int(mask_reg.sum()); DELTA = NS/N
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def random_mask():
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m = np.zeros(N, bool); m[rng.choice(N, NS, replace=False)] = True
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return m.reshape(NX, NY)
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def H(p):
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# entropy of the absolute deviations (Eq. 4 applies it to histogram differences)
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p = np.abs(p); nz = p > 0
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return -np.sum(p[nz]*np.log(p[nz]))
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def pair_hists(dx, dy, weights=None):
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hr, _ = np.histogram(np.hypot(dx, dy), NBR, (0, RMAX), weights=weights)
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# angles folded to [0, pi): the all-pairs rose diagram is 180-degree symmetric
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ha, _ = np.histogram(np.arctan2(dy, dx) % np.pi, NBA, (0, np.pi), weights=weights)
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return hr/hr.max(), ha/ha.max()
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def dense_pair_hists(nx, ny):
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# all-pairs interval/angle histograms of the dense grid via closed-form pair counts
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DX, DY = np.meshgrid(np.arange(-(nx-1), nx), np.arange(0, ny), indexing='ij')
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keep = (DY > 0) | ((DY == 0) & (DX > 0))
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DX, DY = DX[keep], DY[keep]
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w = (nx - np.abs(DX))*(ny - np.abs(DY))
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return pair_hists(DX, DY, w)
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def sparse_pair_hists(mask):
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c = np.argwhere(mask).astype(float)
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d = c[:, None, :] - c[None, :, :]
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iu = np.triu_indices(len(c), 1)
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return pair_hists(d[..., 0][iu], d[..., 1][iu])
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def block_density(mask):
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# per-block occupancy fraction gamma_phi; compared against delta*gamma_theta = DELTA (Eq. 2)
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return mask.reshape(NX//BLK, BLK, NY//BLK, BLK).sum((1, 3))/BLK**2
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def srf(mask):
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# spatial response function of the pattern: mu = largest side lobe (the mutual coherence),
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# xi = the full side-lobe vector
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P = np.abs(np.fft.fft2(mask.astype(float))); P /= P[0, 0]
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xi = P.ravel()[1:]
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return xi.max(), xi
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HR0, HA0 = dense_pair_hists(NX, NY)
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def objective(mask):
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# Eq. 4 with the paper's 2D weights w_alpha = w_beta = w_gamma = mu
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hr, ha = sparse_pair_hists(mask)
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mu, xi = srf(mask)
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return mu*(H(hr - HR0) + H(ha - HA0) + H(block_density(mask) - DELTA) + H(xi))
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def ergodic_mask(iters=SA_ITERS, T0=0.05):
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m = random_mask(); f = objective(m); best = (f, m.copy())
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for it in range(iters):
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T = T0*(1 - it/iters)
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on = np.flatnonzero(m.ravel()); off = np.flatnonzero(~m.ravel())
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i, j = rng.choice(on), rng.choice(off)
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m.flat[i] = False; m.flat[j] = True
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f2 = objective(m)
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if f2 < f or rng.random() < np.exp(-(f2-f)/max(T, 1e-9)):
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f = f2
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if f < best[0]: best = (f, m.copy())
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else:
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m.flat[i] = True; m.flat[j] = False
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return best[1], best[0]
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# ---------------- 4. transforms and compressive-sensing solvers ----------------
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_, SLICES = pywt.coeffs_to_array(pywt.wavedec2(np.zeros((NX, NY)), WAVELET, mode=MODE, level=LEVELS))
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def Wsym(x): return pywt.coeffs_to_array(pywt.wavedec2(x, WAVELET, mode=MODE, level=LEVELS))[0]
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def Wtsym(c): return pywt.waverec2(pywt.array_to_coeffs(c, SLICES, output_format='wavedec2'), WAVELET, mode=MODE)
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SYM_APP = np.zeros((NX, NY), bool); SYM_APP[SLICES[0]] = True # approximation subband
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def Wfft(x): return np.fft.fft2(x)
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def Wtfft(c): return np.fft.ifft2(c).real
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def cs_reconstruct(y_img, mask, W=Wfft, Wt=Wtfft, protect=None,
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n_iter=POCS_ITERS, lam_min_frac=POCS_LMIN):
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# iterative thresholding + data reinsertion (POCS); coefficients flagged in `protect` are
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# never thresholded and excluded from the threshold schedule
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x = y_img.copy()
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c0 = W(y_img)
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lam_max = np.abs(c0 if protect is None else np.where(protect, 0, c0)).max()
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for lam in np.geomspace(lam_max, lam_max*lam_min_frac, n_iter):
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c = W(x)
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t = c*np.maximum(1 - lam/np.maximum(np.abs(c), 1e-300), 0) # soft threshold on magnitude
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x = Wt(t if protect is None else np.where(protect, c, t))
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x[mask] = y_img[mask]
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return x
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# symlet variant (the paper's transform), tuned to reproduce the paper's accuracy in its regime:
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# never threshold the approximation subband, and average reconstructions over circular shifts
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# (cycle spinning) to remove the decimated transform's shift-variance. For the periodized
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# 2-level transform, shifting data and mask together by any multiple of 4 gives an exactly
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# equivalent problem, so only the shift residues modulo 2^LEVELS matter; the fixed list below
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# covers distinct residue pairs (randomness is not needed and keeps the solver deterministic).
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SYM_SHIFTS = [(0, 0), (1, 2), (2, 1), (3, 3), (0, 2), (1, 1), (2, 3), (3, 0)][:SYM_SPINS]
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def cs_reconstruct_sym(y_img, mask, n_iter=POCS_ITERS, lam_min_frac=POCS_LMIN):
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recs = []
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for s in SYM_SHIFTS:
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xs = cs_reconstruct(np.roll(y_img, s, (0, 1)), np.roll(mask, s, (0, 1)),
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Wsym, Wtsym, SYM_APP, n_iter, lam_min_frac)
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recs.append(np.roll(xs, (-s[0], -s[1]), (0, 1)))
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x = np.mean(recs, 0)
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x[mask] = y_img[mask]
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return x
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# ---------------- 5. kriging / GP: Bayes-optimal for the power-law field class ----------------
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GRID = np.argwhere(np.ones((NX, NY), bool))
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def circulant_cov(beta, z_src):
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c = np.fft.ifft2(power_spectrum(beta, z_src)).real
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return c/c[0, 0]
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def sample_cov(on, cov):
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# sample-sample covariance with a small jitter for numerical stability
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K = cov[(on[:, None, 0]-on[None, :, 0]) % NX, (on[:, None, 1]-on[None, :, 1]) % NY]
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return K + 1e-6*np.eye(len(on))
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def gp_matrices(mask, beta, z_src):
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on = np.argwhere(mask)
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cov = circulant_cov(beta, z_src)
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Kxs = cov[(GRID[:, None, 0]-on[None, :, 0]) % NX, (GRID[:, None, 1]-on[None, :, 1]) % NY]
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return sample_cov(on, cov), Kxs
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def gp_unit_variance(Kss, Kxs):
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# posterior variance at every cell for a unit-variance field
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return np.maximum(1.0 - np.einsum('ij,ji->i', Kxs, np.linalg.solve(Kss, Kxs.T)), 0)
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def gp_fit_params(y_img, mask):
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# truth-free ML over spectral slope and source depth, variance profiled out
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# (only the sample-sample covariance is needed here, not the full-grid cross-covariance)
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yv = y_img[mask]; on = np.argwhere(mask); best = None
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for beta in np.arange(1.5, 4.01, 0.5):
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for z_src in (0.0, 0.5, 1.0, 1.5, 2.0, 3.0):
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Kss = sample_cov(on, circulant_cov(beta, z_src))
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try:
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L = np.linalg.cholesky(Kss)
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except np.linalg.LinAlgError:
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continue
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a = np.linalg.solve(L, yv)
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ll = -0.5*len(yv)*np.log(a @ a/len(yv)) - np.log(np.diag(L)).sum()
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if best is None or ll > best[0]: best = (ll, beta, z_src)
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return best[1], best[2]
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def gp_predictor(mask, beta, z_src):
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# precomputed kriging weights: with the mask and model fixed, reconstruction is a single
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# matrix multiplication per field (the second argument exists only for signature parity)
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Kss, Kxs = gp_matrices(mask, beta, z_src)
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Wm = Kxs @ np.linalg.inv(Kss)
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return lambda y_img, m=None: (Wm @ y_img[mask]).reshape(NX, NY)
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def gp_reconstruct(y_img, mask, beta, z_src):
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yv = y_img[mask]
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Kss, Kxs = gp_matrices(mask, beta, z_src)
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xh = (Kxs @ np.linalg.solve(Kss, yv)).reshape(NX, NY)
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var = gp_unit_variance(Kss, Kxs)
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s2 = yv @ np.linalg.solve(Kss, yv)/len(yv) # profiled variance scale
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return xh, np.sqrt(var*s2).reshape(NX, NY)
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def apriori_level_correction(beta, z_src, mask, n_top=400, m_draws=20000):
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# The potential-field generator uses fixed-amplitude random phases and divides each
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# realization by its sample standard deviation. For a linear reconstruction the resulting
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# error variance is E[X/Y] with X = sum_p t_p u_p (per-mode error contributions) and
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# Y = sum_p s_p u_p (the realization variance), where u_p = 1 + cos(theta_p) are the random
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# mode powers. Both expectations involve only scalar mode powers, so the level bias of the
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# posterior sigma is computable a priori from the fitted spectrum and the mask alone.
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Sn = power_spectrum(beta, z_src)
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Sn = (Sn/Sn.sum()).ravel()
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idx = np.arange(N); ix, iy = np.divmod(idx, NY)
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neg = ((-ix) % NX)*NY + ((-iy) % NY)
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take = ((idx < neg) | (idx == neg)) & (Sn > 0)
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s_pair = np.where(idx == neg, Sn, 2*Sn)[take]
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kxp, kyp = ix[take], iy[take]
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o = np.argsort(s_pair)[::-1]
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s_pair, kxp, kyp = s_pair[o], kxp[o], kyp[o]
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rs = np.random.default_rng(123) # local stream, leaves rng untouched
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u_top = 1 + np.cos(2*np.pi*rs.random((m_draws, n_top)))
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Y = u_top @ s_pair[:n_top]
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for j0 in range(n_top, len(s_pair), 512):
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blk = s_pair[j0:j0+512]
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Y += (1 + np.cos(2*np.pi*rs.random((m_draws, len(blk))))) @ blk
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g_top = (u_top/Y[:, None]).mean(0) # E[u_p / Y] for the dominant modes
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g_tail = (1/Y).mean() # small modes are independent of Y
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Kss_c, Kxs_c = gp_matrices(mask, beta, z_src)
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Wc = Kxs_c @ np.linalg.inv(Kss_c)
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sig2_u = gp_unit_variance(Kss_c, Kxs_c)
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ph = np.exp(2j*np.pi*(np.outer(GRID[:, 0], kxp[:n_top])/NX
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+ np.outer(GRID[:, 1], kyp[:n_top])/NY))
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resp = Wc @ ph[mask.ravel()] - ph # error response of each dominant mode
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t_top = (np.abs(resp)**2)*s_pair[:n_top]/2
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t_tail = np.maximum(sig2_u - t_top.sum(1), 0)
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sig2_c = t_top @ g_top + t_tail*g_tail
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return np.sqrt(sig2_c/np.maximum(sig2_u, 1e-30)).reshape(NX, NY)
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# ---------------- 6. metrics, baselines, and validation helpers ----------------
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def leakage(x, truth): return np.corrcoef((x - truth).ravel(), truth.ravel())[0, 1]
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def rms(x, truth): return np.sqrt(np.mean((x - truth)**2))
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def spline_interp(img, mask):
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pts = np.argwhere(mask); X, Y = np.mgrid[0:NX, 0:NY]
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xr = griddata(pts, img[mask], (X, Y), method='cubic')
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nn = np.isnan(xr)
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xr[nn] = griddata(pts, img[mask], (X, Y), method='nearest')[nn]
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return xr
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def fourier_oracle(truth, k=NS):
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C = np.fft.fft2(truth).ravel()
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C[np.argsort(np.abs(C))[:-k]] = 0
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return np.fft.ifft2(C.reshape(NX, NY)).real
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# LOO calibration: reconstruct without one sample, compare prediction at that point to its value
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def loo_rms(y_img, mask, rec_fn, m=LOO_M):
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on = np.argwhere(mask)
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rs = []
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for i in rng.choice(len(on), m, replace=False):
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m2 = mask.copy(); m2[tuple(on[i])] = False
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rs.append(rec_fn(y_img*m2, m2)[tuple(on[i])] - y_img[tuple(on[i])])
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return np.sqrt(np.mean(np.square(rs)))
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def recalibrate(sig, loo):
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return sig*loo/np.sqrt(np.mean(sig**2))
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# a single realization |err| = sigma*|z| caps the achievable correlation even for a perfect map:
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def pearson_ceiling(s):
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return np.sqrt((2/np.pi)*s.var()/((s**2).mean() - (2/np.pi)*s.mean()**2))
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def spearman_ceiling(s, ndraw=50):
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return np.mean([spearmanr(s.ravel()*np.abs(rng.standard_normal(s.size)), s.ravel())[0]
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for _ in range(ndraw)])
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def reliability(s, a, nb=10, reduce=np.mean):
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q = np.quantile(s, np.linspace(0, 1, nb+1)); q[-1] += 1e-12
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m, r = [], []
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for lo, hi in zip(q[:-1], q[1:]):
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sel = (s >= lo) & (s < hi)
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if sel.any():
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m.append(s[sel].mean()); r.append(reduce(a[sel]))
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return np.array(m), np.array(r)
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def rms_reduce(a): return np.sqrt(np.mean(a**2))
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# ---------------- 7. method registry and figure helpers ----------------
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# every pattern-method combination shown in tables and figures; the key prefix names the
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# sampling pattern, resolved through MASKS once the ergodic mask has been optimized
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METHODS = { # key -> (figure column title, Monte-Carlo bar label)
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'sparse+spline': ('Sparse + spline interpolation\n(naive control)',
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'Sparse + spline interpolation'),
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'sparse+CS-Fourier': ('Sparse + compressive sensing\n(Fourier)',
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'Sparse + compressive sensing (Fourier)'),
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'sparse+CS-symlet': ('Sparse + compressive sensing\n(symlet wavelet, from paper)',
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'Sparse + compressive sensing (symlet wavelet)'),
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'sparse+kriging': ('Sparse + kriging\n(Gaussian process)',
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'Sparse + kriging (Gaussian process)'),
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'ergodic+spline': ('Ergodic + spline interpolation',
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'Ergodic + spline interpolation'),
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'ergodic+CS-Fourier': ('Ergodic + compressive sensing\n(Fourier)',
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'Ergodic + compressive sensing (Fourier)'),
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'ergodic+CS-symlet': ('Ergodic + compressive sensing\n(symlet wavelet, from paper)',
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'Ergodic + compressive sensing (symlet wavelet)'),
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'ergodic+kriging': ('Ergodic + kriging\n(Gaussian process)',
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'Ergodic + kriging (Gaussian process)'),
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}
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MC_KEYS = [k for k in METHODS if k != 'sparse+CS-symlet'] # Monte-Carlo method set
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def run_methods(truth, krig_sparse, krig_erg, keys=None):
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# all reconstructions of one truth; the kriging callables are supplied by the caller
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# (gp_reconstruct closures in the tier sections, precomputed weights in the ensembles)
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out = {}
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for k in (keys or list(METHODS)):
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pattern, method = k.split('+')
|
||
m = MASKS[pattern]; y = truth*m
|
||
if method == 'spline': out[k] = spline_interp(y, m)
|
||
elif method == 'CS-Fourier': out[k] = cs_reconstruct(y, m)
|
||
elif method == 'CS-symlet': out[k] = cs_reconstruct_sym(y, m)
|
||
else: out[k] = (krig_sparse if pattern == 'sparse' else krig_erg)(y, m)
|
||
return out
|
||
|
||
# plot colors (figure text carries no abbreviations, a standing convention of this project)
|
||
C_ENS, C_ONE = '#1f77b4', '#ff7f0e' # blue = ensemble comparison, orange = single realization
|
||
C_LIGHT_BLUE, C_LIGHT_ORANGE = '#9ecae1', '#fdbe85' # scatter points / pattern bar colors
|
||
|
||
def plot_mask_panel(a, mm, title):
|
||
a.scatter(*np.argwhere(mm).T, s=2, c='k')
|
||
a.set_aspect(1); a.set_xlim(-1, NX); a.set_ylim(-1, NY)
|
||
a.set_title(title, fontsize=10)
|
||
a.set_xlabel('grid x (cells)')
|
||
|
||
# --- figures 2a1/2b1 (kriging) and 2a2/2b2 (symlet compressive sensing): predicted standard
|
||
# --- deviation versus actual error, one figure per ground truth and reconstruction method ---
|
||
def sigma_study(fname, desc, make_truth, truth, x_rec, sig_raw, rec_fn, rec_name, corr=None,
|
||
ens_rec_fn=None):
|
||
y_img = truth*mask_erg
|
||
loo = loo_rms(y_img, mask_erg, rec_fn)
|
||
if corr is None:
|
||
sig = recalibrate(sig_raw, loo); level_note = 'sigma level set by leave-one-out'
|
||
else:
|
||
sig = sig_raw*corr; level_note = 'sigma level corrected a priori'
|
||
err = x_rec - truth; aerr = np.abs(err)
|
||
print(f"\n--- sigma study ({fname}): {desc} ---")
|
||
print(f"LOO error rms (truth-free check): {loo:.3f} "
|
||
f"actual rms: {np.sqrt((err**2).mean()):.3f} ({level_note})")
|
||
pr = pearsonr(aerr.ravel(), sig.ravel())[0]; sr = spearmanr(aerr.ravel(), sig.ravel())[0]
|
||
pc, sc = pearson_ceiling(sig), spearman_ceiling(sig)
|
||
print(f"|err| vs sigma_GP: pearson={pr:+.3f} (ceiling {pc:.3f}, ratio {pr/pc:+.2f}) "
|
||
f"spearman={sr:+.3f} (ceiling {sc:.3f}, ratio {sr/sc:+.2f}) "
|
||
f"z-std={np.std(err[OFF]/sig[OFF]):.2f}")
|
||
ens_rec = ens_rec_fn if ens_rec_fn is not None else rec_fn
|
||
ens_errs = np.array([ens_rec(t*mask_erg, mask_erg) - t
|
||
for t in (make_truth() for _ in range(ENS_K))])
|
||
sig_emp = np.sqrt((ens_errs**2).mean(0))
|
||
pr2 = pearsonr(sig_emp.ravel(), sig.ravel())[0]; sr2 = spearmanr(sig_emp.ravel(), sig.ravel())[0]
|
||
print(f"ensemble (K={ENS_K} truths, same mask; sigma_emp has ~{1/np.sqrt(2*ENS_K):.0%} noise): "
|
||
f"pearson={pr2:+.3f} spearman={sr2:+.3f} "
|
||
f"pooled z-std={np.std(ens_errs[:, OFF]/sig[OFF]):.2f}")
|
||
|
||
fig, axes = plt.subplots(3, 3, figsize=(14.5, 13.8), constrained_layout=True)
|
||
# fixed axes (3.0 truth standard deviations) so the plt2 figures are directly comparable;
|
||
# some values clip by design
|
||
v = 3.0
|
||
im0 = axes[0, 0].imshow(truth.T, origin='lower', vmin=-v, vmax=v, cmap='RdBu_r')
|
||
axes[0, 0].set_title('Ground truth', fontsize=10)
|
||
axes[0, 1].imshow(x_rec.T, origin='lower', vmin=-v, vmax=v, cmap='RdBu_r')
|
||
axes[0, 1].set_title(f'Reconstruction: ergodic + {rec_name}\n'
|
||
f'leakage {leakage(x_rec, truth):+.2f} · RMS error '
|
||
f'{rms(x_rec, truth):.3f}', fontsize=10)
|
||
axes[0, 2].imshow(err.T, origin='lower', vmin=-v, vmax=v, cmap='RdBu_r')
|
||
axes[0, 2].set_title('Signed error:\nreconstruction - truth', fontsize=10)
|
||
fig.colorbar(im0, ax=axes[0], shrink=0.9,
|
||
label='field value or error\n(units of the truth standard deviation)')
|
||
for a, x, t in zip(axes[1],
|
||
[sig, sig_emp, aerr],
|
||
['Predicted standard deviation:\nGaussian process posterior (truth-free)',
|
||
f'Empirical standard deviation\n({ENS_K} independent truths, '
|
||
'same sampling pattern)',
|
||
'Actual absolute error, this realization:\n|reconstruction - truth|']):
|
||
im1 = a.imshow(x.T, origin='lower', vmin=0, vmax=v, cmap='turbo')
|
||
a.set_title(t, fontsize=10)
|
||
a.set_xlabel('grid x (cells)')
|
||
fig.colorbar(im1, ax=axes[1], shrink=0.9,
|
||
label='standard deviation or absolute error\n(units of the truth standard deviation)')
|
||
for r in (0, 1):
|
||
axes[r, 0].set_ylabel('grid y (cells)')
|
||
|
||
# the sampling pattern that produced everything above
|
||
plot_mask_panel(axes[2, 0], mask_erg, f'Ergodic sampling locations\n({NS} of {N} samples)')
|
||
axes[2, 0].set_ylabel('grid y (cells)')
|
||
|
||
a = axes[2, 1] # scatter with decile-binned means: does the prediction rank the error?
|
||
a.scatter(sig[OFF], aerr[OFF], s=2, alpha=.2, color=C_LIGHT_BLUE)
|
||
bm, bv = reliability(sig[OFF], aerr[OFF])
|
||
a.plot(bm, bv, 'o-', color='k', lw=1.6)
|
||
xs = np.linspace(0, sig[OFF].max(), 50)
|
||
a.plot(xs, np.sqrt(2/np.pi)*xs, '--', color='k', lw=1.2)
|
||
a.set_xlabel('predicted standard deviation'); a.set_ylabel('absolute error, this realization')
|
||
a.set_ylim(0, v)
|
||
a.ticklabel_format(style='sci', scilimits=(-2, 3))
|
||
a.set_title('Prediction versus error at unsampled cells\n(single-realization Pearson '
|
||
f'correlation is capped\nat about {pc:.2f} even for a perfect prediction)',
|
||
fontsize=10)
|
||
a.legend(handles=[plt.Line2D([], [], marker='o', ls='none', ms=4, color=C_LIGHT_BLUE,
|
||
label='unsampled cells'),
|
||
plt.Line2D([], [], marker='o', ls='-', ms=5, color='k',
|
||
label='mean absolute error\nwithin prediction decile'),
|
||
plt.Line2D([], [], ls='--', color='k',
|
||
label='expected mean absolute error\nfor a perfect prediction')],
|
||
loc='upper left', fontsize=8)
|
||
|
||
a = axes[2, 2] # merged calibration panel: against one realization and against the ensemble
|
||
for target, lab, cc in [(aerr, 'against this single realization', C_ONE),
|
||
(sig_emp, f'against the {ENS_K}-truth ensemble', C_ENS)]:
|
||
mm, rr = reliability(sig, target, reduce=rms_reduce)
|
||
a.plot(mm, rr, 'o-', color=cc, label=lab)
|
||
a.plot([0, 1], [0, 1], 'k--', lw=.8, label='perfect calibration')
|
||
a.set_xlim(0, 1.0); a.set_ylim(0, 1.0); a.set_aspect(1)
|
||
a.set_xlabel('mean predicted standard deviation within decile')
|
||
a.set_ylabel('RMS error or\nempirical standard deviation within decile')
|
||
a.ticklabel_format(style='sci', scilimits=(-2, 3))
|
||
a.set_title('Calibration of the Gaussian process prediction\n(cells binned into deciles of '
|
||
'the prediction)', fontsize=10)
|
||
a.legend(loc='upper left', fontsize=8)
|
||
fig.suptitle('Statistical analysis of reconstruction error\n'
|
||
f'({desc}; ergodic sampling; {rec_name} reconstruction)', fontsize=11)
|
||
plt.savefig(f'./{fname}.png', dpi=120)
|
||
|
||
# ---------------- 8. sanity check: is the Gaussian process sigma statistically valid? ----------------
|
||
# With the mask fixed, kriging is a linear map, so a large ensemble is cheap: precompute the
|
||
# weight matrix once and run SANITY_K fresh truths through it. Under the hypothesis that the
|
||
# posterior standard deviation is correct, the empirical standard deviation over K trials should
|
||
# differ from it only by sampling noise (relative size about 1/sqrt(2K) per cell). The unsampled
|
||
# cells are spatially correlated within a single realization, so the honest global test averages
|
||
# the normalized squared error within each trial and treats the SANITY_K per-trial means as the
|
||
# independent observations.
|
||
def run_sanity_check():
|
||
from scipy.stats import norm as normal_dist
|
||
|
||
SANITY_K = 512
|
||
Kss_s, Kxs_s = gp_matrices(mask_erg, BETA_HAT, Z_HAT)
|
||
W_krig = Kxs_s @ np.linalg.inv(Kss_s)
|
||
sig_pred_unit = np.sqrt(gp_unit_variance(Kss_s, Kxs_s)).reshape(NX, NY) # unit-variance field
|
||
sanity_errs = np.empty((SANITY_K, NX, NY))
|
||
for k in range(SANITY_K):
|
||
t = powerlaw_field(BETA)
|
||
sanity_errs[k] = (W_krig @ t[mask_erg]).reshape(NX, NY) - t
|
||
sig_emp_big = np.sqrt((sanity_errs**2).mean(0))
|
||
resid = sig_emp_big - sig_pred_unit
|
||
|
||
# per-trial mean of error^2/sigma^2 over unsampled cells: one independent number per trial
|
||
m_k = (sanity_errs[:, OFF]**2/np.maximum(sig_pred_unit[OFF]**2, 1e-12)).mean(1)
|
||
z_glob = (m_k.mean() - 1)/(m_k.std(ddof=1)/np.sqrt(SANITY_K))
|
||
p_glob = 2*normal_dist.sf(abs(z_glob))
|
||
ratio = sig_emp_big[OFF]/sig_pred_unit[OFF]
|
||
print(f"\n--- sanity check: Gaussian process sigma vs {SANITY_K}-trial empirical sigma ---")
|
||
print(f"mean of per-trial normalized squared error: {m_k.mean():.4f} (1.0 if sigma is correct)")
|
||
print(f"global test over {SANITY_K} independent trials: z = {z_glob:+.2f}, "
|
||
f"two-sided p = {p_glob:.3f}")
|
||
print(f"per-cell ratio sigma_empirical/sigma_predicted at {OFF.sum()} unsampled cells: "
|
||
f"mean {ratio.mean():.4f}, spread {ratio.std():.4f} "
|
||
f"(pure sampling noise predicts spread ~ {1/np.sqrt(2*SANITY_K):.4f})")
|
||
corr_sanity = pearsonr(sig_pred_unit[OFF], sig_emp_big[OFF])[0]
|
||
print(f"correlation(sigma_predicted, sigma_empirical) = {corr_sanity:.4f}")
|
||
|
||
# control: truths drawn exactly from the fitted Gaussian model with a deterministic scale. The
|
||
# pipeline generator uses fixed-amplitude random phases and normalizes each realization by its
|
||
# sample standard deviation, which perturbs the ensemble covariance away from the model; the
|
||
# exact-model truths isolate the question "is the posterior itself constructed correctly?"
|
||
SQ_S = np.sqrt(power_spectrum(BETA_HAT, Z_HAT))
|
||
C0_S = np.fft.ifft2(SQ_S**2).real[0, 0]
|
||
ctrl_errs = np.empty((SANITY_K, NX, NY))
|
||
for k in range(SANITY_K):
|
||
t = np.fft.ifft2(SQ_S*np.fft.fft2(rng.standard_normal((NX, NY)))).real/np.sqrt(C0_S)
|
||
ctrl_errs[k] = (W_krig @ t[mask_erg]).reshape(NX, NY) - t
|
||
m2_k = (ctrl_errs[:, OFF]**2/np.maximum(sig_pred_unit[OFF]**2, 1e-12)).mean(1)
|
||
z_ctrl = (m2_k.mean() - 1)/(m2_k.std(ddof=1)/np.sqrt(SANITY_K))
|
||
p_ctrl = 2*normal_dist.sf(abs(z_ctrl))
|
||
ratio_ctrl = np.sqrt((ctrl_errs**2).mean(0))[OFF]/sig_pred_unit[OFF]
|
||
print(f"control with exact-model truths (no per-realization normalization): "
|
||
f"mean ratio {m2_k.mean():.4f}, z = {z_ctrl:+.2f}, two-sided p = {p_ctrl:.3f}")
|
||
|
||
# same pipeline ensemble scored against the a-priori corrected sigma
|
||
sig_corr_unit = sig_pred_unit*CORR_PL
|
||
m3_k = (sanity_errs[:, OFF]**2/np.maximum(sig_corr_unit[OFF]**2, 1e-12)).mean(1)
|
||
z_corr = (m3_k.mean() - 1)/(m3_k.std(ddof=1)/np.sqrt(SANITY_K))
|
||
p_corr = 2*normal_dist.sf(abs(z_corr))
|
||
ratio_corr = np.sqrt((sanity_errs**2).mean(0))[OFF]/sig_corr_unit[OFF]
|
||
print(f"pipeline truths with the a-priori level correction: "
|
||
f"mean ratio {m3_k.mean():.4f}, z = {z_corr:+.2f}, two-sided p = {p_corr:.3f}")
|
||
|
||
fig, axes = plt.subplots(1, 3, figsize=(16.5, 4.9), constrained_layout=True)
|
||
vr = np.abs(resid).max()
|
||
imR = axes[0].imshow(np.where(OFF, resid, np.nan).T, origin='lower', vmin=-vr, vmax=vr,
|
||
cmap='RdBu_r')
|
||
axes[0].set_title(f'Empirical ({SANITY_K} trials) − predicted standard deviation\n'
|
||
'(sampled cells blanked; structure here would mean a construction error)',
|
||
fontsize=10)
|
||
axes[0].set_xlabel('grid x (cells)'); axes[0].set_ylabel('grid y (cells)')
|
||
fig.colorbar(imR, ax=axes[0], shrink=0.9,
|
||
label='difference\n(units of the truth standard deviation)')
|
||
a = axes[1]
|
||
a.scatter(sig_pred_unit[OFF], sig_emp_big[OFF], s=2, alpha=.25, color=C_LIGHT_BLUE)
|
||
lim = max(sig_pred_unit[OFF].max(), sig_emp_big[OFF].max())*1.05
|
||
a.plot([0, lim], [0, lim], 'k--', lw=1, label='equality')
|
||
a.set_xlim(0, lim); a.set_ylim(0, lim); a.set_aspect(1)
|
||
a.set_xlabel('predicted standard deviation')
|
||
a.set_ylabel(f'empirical standard deviation ({SANITY_K} trials)')
|
||
a.set_title('Per-cell agreement at unsampled cells\n'
|
||
f'(correlation {corr_sanity:.3f})', fontsize=10)
|
||
a.legend(loc='upper left', fontsize=8)
|
||
a = axes[2]
|
||
zn = np.sqrt(2*SANITY_K)*(ratio - 1) # approximately standard normal if sigma is correct
|
||
zn_ctrl = np.sqrt(2*SANITY_K)*(ratio_ctrl - 1)
|
||
zn_corr = np.sqrt(2*SANITY_K)*(ratio_corr - 1)
|
||
a.hist(zn, bins=50, density=True, color=C_LIGHT_BLUE, label='pipeline truths')
|
||
a.hist(zn_corr, bins=50, density=True, histtype='step', color='#2ca02c', lw=1.5,
|
||
label='pipeline truths,\na-priori corrected sigma')
|
||
a.hist(zn_ctrl, bins=50, density=True, histtype='step', color=C_ONE, lw=1.5,
|
||
label='exact-model truths (control)')
|
||
xs = np.linspace(min(zn_corr.min(), -4.5), max(zn.max(), 4.5), 200)
|
||
a.plot(xs, normal_dist.pdf(xs), 'k-', lw=1.2, label='expected from sampling\nnoise alone')
|
||
a.set_xlabel('normalized per-cell deviation of the ratio\nempirical/predicted standard deviation')
|
||
a.set_ylabel('probability density')
|
||
a.set_title(f'Deviations vs pure sampling noise\n(pipeline: z = {z_glob:+.2f}; corrected: '
|
||
f'z = {z_corr:+.2f}; control: z = {z_ctrl:+.2f})', fontsize=10)
|
||
a.legend(loc='upper right', fontsize=8)
|
||
fig.suptitle('Validity check of the Gaussian process standard deviation over '
|
||
f'{SANITY_K} independent truths: shape agreement {corr_sanity:.3f}.\n'
|
||
'The level offset for pipeline truths comes from their per-realization '
|
||
'normalization, not from the posterior: it is computable a priori from the '
|
||
'spectrum and mask,\nand correcting for it centers the test '
|
||
f'(z = {z_corr:+.2f}, p = {p_corr:.2f}); truths drawn exactly from the Gaussian '
|
||
f'model also pass (z = {z_ctrl:+.2f}, p = {p_ctrl:.2f}).', fontsize=11)
|
||
plt.savefig('./plt4.png', dpi=120)
|
||
|
||
# ================ EXPERIMENT (everything below acquires, reconstructs, and plots) ================
|
||
|
||
# --- ground truths and sampling patterns ---
|
||
truth_pl = powerlaw_field(BETA) # tier 2: hard, not l1-sparse in any basis
|
||
truth_sp = fourier_sparse_field(N_MODES) # tier 1: exactly sparse, the paper's Fig. A.1 setting
|
||
|
||
mask_erg, f_erg = ergodic_mask()
|
||
MASKS = {'sparse': mask_reg, 'ergodic': mask_erg}
|
||
OFF = ~mask_erg # evaluate z-scores at unsampled pixels only (err and sigma are both 0 at samples)
|
||
f_rand = np.mean([objective(random_mask()) for _ in range(20)])
|
||
print(f"N_samples={NS} (delta={DELTA:.3f}) obj: sparse={objective(mask_reg):.3f} "
|
||
f"random(mean)={f_rand:.3f} ergodic={f_erg:.3f}")
|
||
|
||
# --- tier 1: Fourier-sparse truth: the CS claim of the paper (Fig. A.1 / Figs. 15, 22) ---
|
||
print("\n--- tier 1: Fourier-sparse truth (paper's CS regime) ---")
|
||
# kriging with the power-law covariance is misspecified for a 12-mode signal; shown as the
|
||
# symmetric counterpart to CS-on-the-random-field so every process appears on both truths
|
||
GP_SP = gp_fit_params(truth_sp*mask_erg, mask_erg)
|
||
krig_sp = lambda y, m: gp_reconstruct(y, m, *GP_SP)[0]
|
||
recs_sp = run_methods(truth_sp, krig_sp, krig_sp)
|
||
sig_sp_raw = gp_reconstruct(truth_sp*mask_erg, mask_erg, *GP_SP)[1]
|
||
for name, x in recs_sp.items():
|
||
print(f"{name:18s} leakage={leakage(x, truth_sp):+.3f} rms={rms(x, truth_sp):.3f}")
|
||
|
||
# --- tier 2: potential-field truth: not l1-sparse; report against the oracle floor ---
|
||
print(f"\n--- tier 2: potential-field truth (beta={BETA}, source depth {Z_SRC}) ---")
|
||
print(f"oracle {NS}-term Fourier floor: rms={rms(fourier_oracle(truth_pl), truth_pl):.3f}")
|
||
BETA_HAT, Z_HAT = gp_fit_params(truth_pl*mask_erg, mask_erg)
|
||
print(f"GP ML estimates: beta_hat={BETA_HAT:.1f} z_hat={Z_HAT:.1f} "
|
||
f"(generator beta={BETA}, z={Z_SRC})")
|
||
CORR_PL = apriori_level_correction(BETA_HAT, Z_HAT, mask_erg)
|
||
print(f"a-priori sigma level correction for the normalized truth generator: "
|
||
f"x{CORR_PL[~mask_erg].mean():.4f} (spread {CORR_PL[~mask_erg].std():.4f})")
|
||
krig_pl = lambda y, m: gp_reconstruct(y, m, BETA_HAT, Z_HAT)[0]
|
||
recs_pl = run_methods(truth_pl, krig_pl, krig_pl)
|
||
sig_gp_raw = gp_reconstruct(truth_pl*mask_erg, mask_erg, BETA_HAT, Z_HAT)[1]
|
||
print("method leakage rms -rms/std(truth) (leakage ~ -rms/std when error is unrecovered signal)")
|
||
for name, x in recs_pl.items():
|
||
print(f"{name:18s} {leakage(x, truth_pl):+.3f} {rms(x, truth_pl):.3f} {-rms(x, truth_pl)/truth_pl.std():+.3f}")
|
||
|
||
# --- figures 1a/1b: one figure per ground truth: reconstructions, errors, sampling locations ---
|
||
COL_TITLES = ['Ground truth'] + [METHODS[k][0] for k in METHODS]
|
||
COL_MASKS = [np.ones_like(mask_reg)] + [MASKS[k.split('+')[0]] for k in METHODS]
|
||
for fname, gt, recs, note in [
|
||
('plt1a', truth_sp, recs_sp,
|
||
f'Fourier-sparse truth ({N_MODES} modes); compressive sensing with the ergodic pattern '
|
||
'recovers the signal exactly'),
|
||
('plt1b', truth_pl, recs_pl,
|
||
f'potential-field truth (spectral exponent {BETA}, source depth {Z_SRC} cells); '
|
||
'kriging is the best reconstruction for this field class')]:
|
||
imgs = [gt] + [recs[k] for k in METHODS]
|
||
fig, axes = plt.subplots(3, 9, figsize=(36, 12.4), constrained_layout=True)
|
||
v = np.abs(gt).max()
|
||
for c, (a, x, t) in enumerate(zip(axes[0], imgs, COL_TITLES)):
|
||
im = a.imshow(x.T, origin='lower', vmin=-v, vmax=v, cmap='RdBu_r')
|
||
sub = '' if c == 0 else (f'\nleakage {leakage(x, gt):+.2f} · '
|
||
f'RMS error {rms(x, gt):.3f}')
|
||
a.set_title(t + sub, fontsize=9)
|
||
axes[0, 0].set_ylabel('Reconstruction\n\ngrid y (cells)')
|
||
fig.colorbar(im, ax=axes[0], shrink=0.9,
|
||
label='field value\n(units of the truth standard deviation)')
|
||
errors = [gt - x for x in imgs[1:]]
|
||
ve = max(np.abs(e).max() for e in errors)
|
||
axes[1, 0].axis('off')
|
||
for a, e in zip(axes[1, 1:], errors):
|
||
im2 = a.imshow(e.T, origin='lower', vmin=-ve, vmax=ve, cmap='RdBu_r')
|
||
axes[1, 1].set_ylabel('Error: ground truth − reconstruction\n\ngrid y (cells)')
|
||
fig.colorbar(im2, ax=axes[1, 1:], shrink=0.9,
|
||
label='error\n(units of the truth standard deviation)')
|
||
for a, mm in zip(axes[2], COL_MASKS):
|
||
plot_mask_panel(a, mm, f'{int(mm.sum())} of {N} samples')
|
||
axes[2, 0].set_ylabel('Sampling locations\n\ngrid y (cells)')
|
||
fig.suptitle(f'Reconstruction from {DELTA:.1%} of the samples: {note}')
|
||
plt.savefig(f'./{fname}.png', dpi=120)
|
||
|
||
# --- figures 2a/2b: predicted standard deviation versus actual error ---
|
||
DESC_SP = f'Fourier-sparse truth ({N_MODES} modes)'
|
||
DESC_PL = f'potential-field truth (spectral exponent {BETA}, source depth {Z_SRC} cells)'
|
||
REC_GP, REC_SYM = 'kriging (Gaussian process)', 'compressive sensing (symlet wavelet, from paper)'
|
||
KRIG_FAST_SP = gp_predictor(mask_erg, *GP_SP)
|
||
KRIG_FAST_PL = gp_predictor(mask_erg, BETA_HAT, Z_HAT)
|
||
sigma_study('plt2a1', DESC_SP, lambda: fourier_sparse_field(N_MODES), truth_sp,
|
||
recs_sp['ergodic+kriging'], sig_sp_raw, krig_sp, REC_GP, ens_rec_fn=KRIG_FAST_SP)
|
||
sigma_study('plt2a2', DESC_SP, lambda: fourier_sparse_field(N_MODES), truth_sp,
|
||
recs_sp['ergodic+CS-symlet'], sig_sp_raw, cs_reconstruct_sym, REC_SYM)
|
||
sigma_study('plt2b1', DESC_PL, lambda: powerlaw_field(BETA), truth_pl,
|
||
recs_pl['ergodic+kriging'], sig_gp_raw, krig_pl, REC_GP, corr=CORR_PL,
|
||
ens_rec_fn=KRIG_FAST_PL)
|
||
sigma_study('plt2b2', DESC_PL, lambda: powerlaw_field(BETA), truth_pl,
|
||
recs_pl['ergodic+CS-symlet'], sig_gp_raw, cs_reconstruct_sym, REC_SYM)
|
||
|
||
# --- Monte-Carlo RMS table over truth classes ---
|
||
CLASSES = [('white noise (control)', white_noise_field),
|
||
('Fourier-sparse', lambda: fourier_sparse_field(N_MODES)),
|
||
('potential-field', lambda: powerlaw_field(BETA))]
|
||
print(f"\n--- Monte-Carlo rms error, mean +/- std over {N_TRIALS} trials per truth class ---")
|
||
results = {}
|
||
for cname, gen in CLASSES:
|
||
t0 = gen()
|
||
gp_params = gp_fit_params(t0*mask_erg, mask_erg) # truth-free model selection, once per class
|
||
krig_sparse = gp_predictor(mask_reg, *gp_params) # weights precomputed once per class
|
||
krig_erg = gp_predictor(mask_erg, *gp_params)
|
||
rr = {}
|
||
for k in range(N_TRIALS):
|
||
truth = t0 if k == 0 else gen()
|
||
for m, x in run_methods(truth, krig_sparse, krig_erg, MC_KEYS).items():
|
||
rr.setdefault(m, []).append(rms(x, truth))
|
||
results[cname] = {m: (np.mean(v), np.std(v)) for m, v in rr.items()}
|
||
print('method'.ljust(20) + ''.join(c.rjust(24) for c, _ in CLASSES))
|
||
for m in MC_KEYS:
|
||
print(m.ljust(20) + ''.join(f"{results[c][m][0]:.3f} +/- {results[c][m][1]:.3f}".rjust(24)
|
||
for c, _ in CLASSES))
|
||
|
||
# --- figure 3: the Monte-Carlo table as a chart with error bars ---
|
||
ypos = np.arange(len(MC_KEYS))[::-1]
|
||
xmax = max(results[c][m][0] + results[c][m][1] for c, _ in CLASSES for m in MC_KEYS)
|
||
fig, axes = plt.subplots(1, 3, figsize=(15.5, 5.8), sharey=True, constrained_layout=True)
|
||
for a, (cname, _) in zip(axes, CLASSES):
|
||
mus = np.array([results[cname][m][0] for m in MC_KEYS])
|
||
sds = np.array([results[cname][m][1] for m in MC_KEYS])
|
||
# bar color encodes the sampling pattern (blue = sparse grid, orange = ergodic); the method
|
||
# labels already carry this, so no legend is needed
|
||
a.barh(ypos, mus, xerr=sds, height=0.6,
|
||
color=[C_LIGHT_BLUE if m.startswith('sparse') else C_LIGHT_ORANGE for m in MC_KEYS],
|
||
error_kw=dict(ecolor='k', lw=1, capsize=3))
|
||
for yp, mu, sd in zip(ypos, mus, sds):
|
||
a.text(mu + sd + 0.02*xmax, yp, f'{mu:.3f} ± {sd:.3f}', va='center', fontsize=8)
|
||
t = cname[0].upper() + cname[1:]
|
||
if not cname.endswith(')'):
|
||
t += ' truth'
|
||
if cname == CLASSES[0][0]:
|
||
t += f'\n(dashed line: best possible = predict zero everywhere, {np.sqrt(1 - DELTA):.2f})'
|
||
a.set_title(t, fontsize=10)
|
||
a.set_xlabel('RMS error\n(units of the truth standard deviation)')
|
||
a.set_xlim(0, xmax*1.32)
|
||
a.grid(axis='x', lw=0.4, alpha=0.4)
|
||
a.set_axisbelow(True)
|
||
axes[0].set_yticks(ypos, [METHODS[m][1] for m in MC_KEYS])
|
||
axes[0].axvline(np.sqrt(1 - DELTA), ls='--', lw=1, color='k')
|
||
fig.suptitle('Monte-Carlo RMS reconstruction error: '
|
||
f'mean ± standard deviation over {N_TRIALS} trials per truth class '
|
||
f'({DELTA:.1%} of samples, error bars = ± one standard deviation)')
|
||
plt.savefig('./plt3.png', dpi=120)
|
||
|
||
if RUN_SANITY_CHECK:
|
||
run_sanity_check()
|