libcov.py 35.9 KB
 Matthieu Tristram committed Jan 14, 2019 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ``````""" Set of routines to ... Author: Vanneste """ from __future__ import division import timeit import numpy as np import healpy as hp import math from scipy import special `````` Matthieu Tristram committed Feb 27, 2019 15 ``````from .simulation import extrapolpixwin `````` Matthieu Tristram committed Mar 14, 2019 16 ``````from .xqml_utils import getstokes, progress_bar, GetBinningMatrix `````` Matthieu Tristram committed Jan 14, 2019 17 18 19 `````` import _libcov as clibcov `````` Matthieu Tristram committed Jan 14, 2019 20 21 `````` def compute_ds_dcb( ellbins, nside, ipok, bl, clth, Slmax, spec, `````` Matthieu Tristram committed Jan 21, 2020 22 `````` pixwin=True, timing=False, MC=0, Sonly=False, openMP=True): `````` Matthieu Tristram committed Jan 14, 2019 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 `````` """ Compute the Pl = dS/dCl matrices. Parameters ---------- ellbins : array of floats Lowers bounds of bins. Example : taking ellbins = (2, 3, 5) will compute the spectra for bins = (2, 3.5). nside : int Healpix map resolution ipok : array of ints Healpy pixels numbers considered bl : 1D array of floats Beam window function clth : 4D or 6D array of float Fiducial power spectra Slmax : int Maximum lmax computed for the pixel covariance pixel matrix spec : 1D array of string Spectra list pixwin : bool If True, multiplies the beam window function by the pixel window function. Default: True timing : bool If True, displays timmer. Default: False MC : int If not 0, computes Pl using Monte-Carlo method from MC simulations. Default: False Sonly : bool If True, compute the signal matric only. Default: False Returns ---------- Pl : ndarray of floats Normalize Legendre polynomials dS/dCl S : 2D array of floats Pixel signal covariance matrix S Example ---------- >>> Pl, S = compute_ds_dcb( `````` Matthieu Tristram committed Feb 27, 2019 66 67 `````` ... np.array([2,4,5,10]), 2, np.array([0,1,4,10,11]), np.arange(10), ... clth=np.arange(60).reshape(6,-1), Slmax=9, `````` Matthieu Tristram committed Jan 14, 2019 68 69 70 `````` ... spec=["TT", "EE", "BB", "TE", "EB", "TB"], ... pixwin=True, timing=False, MC=0, Sonly=False) >>> print(round(np.sum(Pl),5), round(np.sum(S),5)) `````` Matthieu Tristram committed Feb 27, 2019 71 `````` (1149.18805, 23675.10206) `````` Matthieu Tristram committed Jan 14, 2019 72 73 74 75 76 77 78 79 80 81 82 83 `````` """ if Slmax < ellbins[-1]-1: print("WARNING : Slmax < lmax") # ### define pixels rpix = np.array(hp.pix2vec(nside, ipok)) allcosang = np.dot(np.transpose(rpix), rpix) allcosang[allcosang > 1] = 1.0 allcosang[allcosang < -1] = -1.0 start = timeit.default_timer() `````` Matthieu Tristram committed Jan 21, 2020 84 85 86 87 88 `````` clth = np.asarray(clth) if len(clth) == 4: clth = np.concatenate((clth,clth[0:2]*0.)) temp = "TT" in spec `````` Matthieu Tristram committed Jan 14, 2019 89 `````` polar = "EE" in spec or "BB" in spec `````` Matthieu Tristram committed Jan 21, 2020 90 `````` corr = "TE" in spec or "TB" in spec or "EB" in spec `````` Matthieu Tristram committed Jan 14, 2019 91 92 93 `````` if Sonly: if MC: S = S_bins_MC( `````` Matthieu Tristram committed Jan 21, 2020 94 `````` ellbins, nside, ipok, allcosang, bl, clth, Slmax, MC, spec, `````` Matthieu Tristram committed Jan 14, 2019 95 96 97 `````` pixwin=pixwin, timing=timing) else: S = compute_S( `````` Matthieu Tristram committed Jan 21, 2020 98 `````` ellbins, nside, ipok, allcosang, bl, clth, Slmax, spec, `````` Matthieu Tristram committed Jan 14, 2019 99 100 `````` pixwin=pixwin, timing=timing) return S `````` Matthieu Tristram committed Jan 21, 2020 101 `````` `````` Matthieu Tristram committed Jan 14, 2019 102 103 104 `````` if MC: Pl, S = covth_bins_MC( ellbins, nside, ipok, allcosang, bl, clth, Slmax, MC, `````` Matthieu Tristram committed Jan 21, 2020 105 `````` spec, pixwin=pixwin, timing=timing) `````` Matthieu Tristram committed Jan 14, 2019 106 `````` elif openMP: `````` Matthieu Tristram committed Feb 27, 2019 107 `````` fpixwin = extrapolpixwin(nside, Slmax, pixwin) `````` Matthieu Tristram committed Jan 14, 2019 108 109 `````` bell = np.array([bl*fpixwin]*4)[:Slmax+1].ravel() stokes, spec, istokes, ispecs = getstokes(spec) `````` Matthieu Tristram committed Jan 21, 2020 110 111 `````` ispec = np.zeros(6,int) ispec[ispecs] = 1 `````` Matthieu Tristram committed Jan 14, 2019 112 113 114 `````` nbins = (len(ellbins)-1)*len(spec) npix = len(ipok)*len(istokes) Pl = np.ndarray( nbins*npix**2) `````` Matthieu Tristram committed Jan 21, 2020 115 `````` clibcov.dSdC( nside, len(istokes), ispec, ellbins, ipok, bell, Pl) `````` Matthieu Tristram committed Jan 14, 2019 116 `````` Pl = Pl.reshape( nbins, npix, npix) `````` Matthieu Tristram committed Feb 27, 2019 117 `````` P, Q, ell, ellval = GetBinningMatrix(ellbins, Slmax) `````` Matthieu Tristram committed Jan 21, 2020 118 `````` S = SignalCovMatrix(Pl,np.array([P.dot(clth[isp,2:Slmax+1]) for isp in ispecs]).ravel()) `````` Matthieu Tristram committed Jan 14, 2019 119 120 121 122 123 `````` else: Pl, S = compute_PlS( ellbins, nside, ipok, allcosang, bl, clth, Slmax, spec=spec, pixwin=pixwin, timing=timing) `````` Matthieu Tristram committed Jan 21, 2020 124 125 126 `````` if timing: print( "Total time (npix=%d): %.1f sec" % (len(ipok),timeit.default_timer()-start)) `````` Matthieu Tristram committed Jan 14, 2019 127 128 129 `````` return Pl, S `````` Matthieu Tristram committed Jan 14, 2019 130 `````` `````` Matthieu Tristram committed Feb 27, 2019 131 `````` `````` Matthieu Tristram committed Mar 14, 2019 132 ``````def SignalCovMatrix(Pl, model): `````` Matthieu Tristram committed Feb 27, 2019 133 134 135 136 137 138 139 140 141 `````` """ Compute correlation matrix S = sum_l Pl*Cl Parameters ---------- clth : ndarray of floats Array containing fiducial CMB spectra (unbinned). """ # Return scalar product btw Pl and the fiducial spectra. `````` Matthieu Tristram committed Mar 14, 2019 142 `````` return np.sum(Pl * model[:, None, None], 0) `````` Matthieu Tristram committed Feb 27, 2019 143 144 145 146 `````` `````` Matthieu Tristram committed Jan 14, 2019 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 ``````def compute_PlS( ellbins, nside, ipok, allcosang, bl, clth, Slmax, spec, pixwin=True, timing=False): """ Computes Legendre polynomes Pl = dS/dCb and signal matrix S. Parameters ---------- ellbins : array of floats Lowers bounds of bins. Example : taking ellbins = (2, 3, 5) will compute the spectra for bins = (2, 3.5). nside : int Healpix map resolution ipok : array of ints Healpy pixels numbers considered bl : 1D array of floats Beam window function clth : 4D or 6D array of float Fiducial power spectra Slmax : int Maximum lmax computed for the pixel covariance pixel matrix spec : 1D array of string Spectra list pixwin : bool If True, multiplies the beam window function by the pixel window function. Default: True timing : bool If True, displays timmer. Default: False Returns ---------- Pl : ndarray of floats Normalize Legendre polynomials dS/dCl S : 2D array of floats Pixel signal covariance matrix S Example ---------- `````` Matthieu Tristram committed Feb 27, 2019 186 `````` >>> Pl, S = compute_PlS(np.array([2,3,4,10]),4,ipok=np.array([0,1,3]), `````` Matthieu Tristram committed Jan 14, 2019 187 `````` ... allcosang=np.linspace(0,1,15).reshape(3,-1), bl=np.arange(13), `````` Matthieu Tristram committed Feb 27, 2019 188 `````` ... clth=np.arange(6*13).reshape(6,-1), Slmax=9, `````` Matthieu Tristram committed Jan 14, 2019 189 190 `````` ... spec=["TT", "EE", "BB", "TE", "EB", "TB"], pixwin=True, timing=False) >>> print(round(np.sum(Pl),5), round(np.sum(S),5)) `````` Matthieu Tristram committed Feb 27, 2019 191 `````` (-429.8591, -17502.8982) `````` Matthieu Tristram committed Jan 14, 2019 192 `````` `````` Matthieu Tristram committed Feb 27, 2019 193 `````` >>> Pl, S = compute_PlS(np.array([2,3,4,10]),4,ipok=np.array([0,1,3]), `````` Matthieu Tristram committed Jan 14, 2019 194 `````` ... allcosang=np.linspace(0,1,15).reshape(3,-1), bl=np.arange(13), `````` Matthieu Tristram committed Feb 27, 2019 195 `````` ... clth=np.arange(6*13).reshape(6,-1), Slmax=9, `````` Matthieu Tristram committed Jan 14, 2019 196 197 `````` ... spec=["TT", "EE", "BB", "TE", "EB", "TB"], pixwin=False, timing=False) >>> print(round(np.sum(Pl),5), round(np.sum(S),5)) `````` Matthieu Tristram committed Feb 27, 2019 198 `````` (-756.35517, -31333.69722) `````` Matthieu Tristram committed Jan 14, 2019 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 `````` """ lmax = ellbins[-1] ell = np.arange(np.min(ellbins), np.max(ellbins) + 1) nbins = len(ellbins) - 1 minell = np.array(ellbins[0:nbins]) maxell = np.array(ellbins[1:nbins+1]) - 1 ellval = (minell + maxell) * 0.5 npi = ipok.size stokes, spec, istokes, ispecs = getstokes(spec) nspec = len(spec) nsto = len(stokes) temp = "TT" in spec polar = "EE" in spec or "BB" in spec TE = 'TE' in spec EB = 'EB' in spec TB = 'TB' in spec te = spec.index('TE') if TE else 0 tb = spec.index('TB') if TB else 0 eb = spec.index('EB') if EB else 0 ponbins = nbins*temp ponpi = npi*temp tenbins = te*nbins tbnbins = tb*nbins ebnbins = eb*nbins rpix = np.array(hp.pix2vec(nside, ipok)) ll = np.arange(Slmax+1) `````` Matthieu Tristram committed Feb 27, 2019 228 `````` fpixwin = extrapolpixwin(nside, Slmax, pixwin) `````` Matthieu Tristram committed Jan 14, 2019 229 230 231 232 233 234 235 236 237 238 239 240 241 242 `````` norm = (2*ll[2:]+1)/(4.*np.pi)*(fpixwin[2:]**2)*(bl[2:Slmax+1]**2) clthn = clth[:, 2: Slmax+1] masks = [] for i in np.arange(nbins): masks.append((ll[2:] >= minell[i]) & (ll[2:] <= maxell[i])) masks = np.array(masks) Pl = np.zeros((nspec*nbins, nsto*npi, nsto*npi)) S = np.zeros((nsto*npi, nsto*npi)) start = timeit.default_timer() for i in np.arange(npi): if timing: `````` Matthieu Tristram committed Jan 21, 2020 243 `````` progress_bar(i, npi) `````` Matthieu Tristram committed Jan 14, 2019 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 `````` for j in np.arange(i, npi): cos_chi = allcosang[i, j] if temp: pl = norm*pl0(cos_chi, Slmax)[2:] elem = np.sum((pl * clthn[0])) S[i, j] = elem S[j, i] = elem for b in np.arange(nbins): elem = np.sum(pl[masks[b]]) Pl[b, i, j] = elem Pl[b, j, i] = elem if polar: ii = i+ponpi jj = j+ponpi cij, sij = polrotangle(rpix[:, i], rpix[:, j]) cji, sji = polrotangle(rpix[:, j], rpix[:, i]) # Tegmark version Q22 = norm * F1l2(cos_chi, Slmax) # # /!\ signe - ! R22 = -norm * F2l2(cos_chi, Slmax) # # Matt version `````` Matthieu Tristram committed Jan 14, 2019 268 269 270 271 `````` # d2p2 = dlss(cos_chi, 2, 2, Slmax) # d2m2 = dlss(cos_chi, 2, -2, Slmax) # Q22 = norm * ( d2p2 + d2m2 )[2:]/2. # R22 = norm * ( d2p2 - d2m2 )[2:]/2. `````` Matthieu Tristram committed Jan 14, 2019 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 `````` if TE or TB: # # Matt version d20 = -dlss(cos_chi, 2, 0, Slmax)[2:] P02 = norm * d20 # P02 = -norm * F1l0(cos_chi, Slmax) elemA = 0 elemB = 0 elemC = 0 elemD = 0 if TE: elemTE = np.sum(P02*clthn[3]) elemA += cji*elemTE elemB -= sji*elemTE elemC += cij*elemTE elemD -= sij*elemTE if TB: elemTB = np.sum(P02*clthn[5]) elemA += sji*elemTB elemB += cji*elemTB elemC += sij*elemTB elemD += cij*elemTB S[i, jj] = elemA S[i, jj+npi] = elemB S[ii, j] = elemC S[ii+npi, j] = elemD S[jj, i] = elemA S[jj+npi, i] = elemB S[j, ii] = elemC S[j, ii+npi] = elemD elem1 = np.sum((cij*cji*Q22 + sij*sji*R22)*clthn[1]) elem2 = np.sum((-cij*sji*Q22 + sij*cji*R22)*clthn[1]) elem3 = np.sum((sij*sji*Q22 + cij*cji*R22)*clthn[1]) elem4 = np.sum((-sij*cji*Q22 + cij*sji*R22)*clthn[1]) elem3 += np.sum((cij*cji*Q22 + sij*sji*R22)*clthn[2]) elem4 -= np.sum((-cij*sji*Q22 + sij*cji*R22)*clthn[2]) elem1 += np.sum((sij*sji*Q22 + cij*cji*R22)*clthn[2]) elem2 -= np.sum((-sij*cji*Q22 + cij*sji*R22)*clthn[2]) if EB: elemEB = np.sum((Q22 - R22)*clthn[4]) elem1 += (cji*sij + sji*cij)*elemEB elem2 += (-sji*sij + cji*cij)*elemEB elem3 += (-sji*cij - cji*sij)*elemEB elem4 += (cji*cij - sji*sij)*elemEB S[ii, jj] = elem1 S[ii, jj+npi] = elem2 S[ii+npi, jj+npi] = elem3 S[ii+npi, jj] = elem4 S[jj, ii] = elem1 S[jj+npi, ii] = elem2 S[jj+npi, ii+npi] = elem3 S[jj, ii+npi] = elem4 for b in np.arange(nbins): elem1 = np.sum(( cij*cji*Q22+sij*sji*R22)[masks[b]]) elem2 = np.sum((-cij*sji*Q22+sij*cji*R22)[masks[b]]) elem3 = np.sum(( sij*sji*Q22+cij*cji*R22)[masks[b]]) elem4 = np.sum((-sij*cji*Q22+cij*sji*R22)[masks[b]]) # # EE ij then ji Pl[ponbins + b, ii, jj ] = elem1 Pl[ponbins + b, ii, jj+npi] = elem2 Pl[ponbins + b, ii+npi, jj+npi] = elem3 Pl[ponbins + b, ii+npi, jj ] = elem4 Pl[ponbins + b, jj, ii ] = elem1 Pl[ponbins + b, jj+npi, ii ] = elem2 Pl[ponbins + b, jj+npi, ii+npi] = elem3 Pl[ponbins + b, jj, ii+npi] = elem4 # # BB ij then ji Pl[ponbins + nbins+b, ii+npi, jj+npi] = elem1 Pl[ponbins + nbins+b, ii+npi, jj ] = -elem2 Pl[ponbins + nbins+b, ii, jj ] = elem3 Pl[ponbins + nbins+b, ii, jj+npi] = -elem4 Pl[ponbins + nbins+b, jj+npi, ii+npi] = elem1 Pl[ponbins + nbins+b, jj, ii+npi] = -elem2 Pl[ponbins + nbins+b, jj, ii ] = elem3 Pl[ponbins + nbins+b, jj+npi, ii ] = -elem4 if TE or TB: elemA = np.sum(cji*P02[masks[b]]) elemB = np.sum(sji*P02[masks[b]]) elemC = np.sum(cij*P02[masks[b]]) elemD = np.sum(sij*P02[masks[b]]) if TE: Pl[tenbins + b, i, jj ] = elemA Pl[tenbins + b, i, jj+npi] = -elemB Pl[tenbins + b, ii, j] = elemC Pl[tenbins + b, ii+npi, j] = -elemD Pl[tenbins + b, jj, i] = elemA Pl[tenbins + b, jj+npi, i] = -elemB Pl[tenbins + b, j, ii] = elemC Pl[tenbins + b, j, ii+npi] = -elemD if TB: Pl[tbnbins + b, i, jj ] = elemB Pl[tbnbins + b, i, jj+npi] = elemA Pl[tbnbins + b, ii, j ] = elemD Pl[tbnbins + b, ii+npi, j] = elemC Pl[tbnbins + b, jj, i] = elemB Pl[tbnbins + b, jj+npi, i] = elemA Pl[tbnbins + b, j, ii] = elemD Pl[tbnbins + b, j, ii+npi] = elemC if EB: Pl[ebnbins+b, ii, jj] = -elem2-elem4 Pl[ebnbins+b, ii, jj+npi] = elem1-elem3 Pl[ebnbins+b, ii+npi, jj+npi] = elem2+elem4 Pl[ebnbins+b, ii+npi, jj] = elem1-elem3 Pl[ebnbins+b, jj, ii] = -elem2-elem4 Pl[ebnbins+b, jj+npi, ii] = elem1-elem3 Pl[ebnbins+b, jj+npi, ii+npi] = elem2+elem4 Pl[ebnbins+b, jj, ii+npi] = elem1-elem3 return Pl, S def compute_S( ellbins, nside, ipok, allcosang, bl, clth, Slmax, spec, pixwin=True, timing=False): """ Computes signal matrix S. Parameters ---------- ellbins : array of floats Lowers bounds of bins. Example : taking ellbins = (2, 3, 5) will compute the spectra for bins = (2, 3.5). nside : int Healpix map resolution ipok : array of ints Healpy pixels numbers considered bl : 1D array of floats Beam window function clth : 4D or 6D array of float Fiducial power spectra Slmax : int Maximum lmax computed for the pixel covariance pixel matrix spec : 1D array of string Spectra list pixwin : bool If True, multiplies the beam window function by the pixel window function. Default: True timing : bool If True, displays timmer. Default: False Returns ---------- S : 2D array of floats Pixel signal covariance matrix S Example ---------- `````` Matthieu Tristram committed Feb 27, 2019 440 441 442 `````` >>> S = compute_S(np.array([2,3,4,10]),4,ipok=np.array([0,1,3]), ... allcosang=np.linspace(0,1,15).reshape(3,-1), bl=np.arange(10), ... clth=np.arange(6*13).reshape(6,-1), Slmax=9, `````` Matthieu Tristram committed Jan 14, 2019 443 444 `````` ... spec=["TT", "EE", "BB", "TE", "EB", "TB"], pixwin=True, timing=False) >>> print(round(np.sum(S),5)) `````` Matthieu Tristram committed Feb 27, 2019 445 `````` -17502.8982 `````` Matthieu Tristram committed Jan 14, 2019 446 `````` `````` Matthieu Tristram committed Feb 27, 2019 447 `````` >>> S = compute_S(np.array([2,3,4,10]),4,ipok=np.array([0,1,3]), `````` Matthieu Tristram committed Jan 14, 2019 448 `````` ... allcosang=np.linspace(0,1,15).reshape(3,-1), bl=np.arange(13), `````` Matthieu Tristram committed Feb 27, 2019 449 `````` ... clth=np.arange(6*13).reshape(6,-1), Slmax=9, `````` Matthieu Tristram committed Jan 14, 2019 450 451 `````` ... spec=["TT", "EE", "BB", "TE", "EB", "TB"], pixwin=False, timing=False) >>> print(round(np.sum(S),5)) `````` Matthieu Tristram committed Feb 27, 2019 452 `````` -31333.69722 `````` Matthieu Tristram committed Jan 14, 2019 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 `````` """ lmax = ellbins[-1] ell = np.arange(np.min(ellbins), np.max(ellbins) + 1) nbins = len(ellbins) - 1 minell = np.array(ellbins[0:nbins]) maxell = np.array(ellbins[1:nbins+1]) - 1 ellval = (minell + maxell) * 0.5 npi = ipok.size stokes, spec, istokes, ispecs = getstokes(spec) nspec = len(spec) nsto = len(stokes) temp = "TT" in spec polar = "EE" in spec or "BB" in spec TE = 'TE' in spec EB = 'EB' in spec TB = 'TB' in spec te = spec.index('TE') if TE else 0 tb = spec.index('TB') if TB else 0 eb = spec.index('EB') if EB else 0 ponbins = nbins*temp ponpi = npi*temp tenbins = te*nbins tbnbins = tb*nbins ebnbins = eb*nbins rpix = np.array(hp.pix2vec(nside, ipok)) ll = np.arange(Slmax+1) `````` Matthieu Tristram committed Feb 27, 2019 482 483 `````` fpixwin = extrapolpixwin(nside, Slmax, pixwin) norm = (2*ll[2:]+1)/(4.*np.pi)*(fpixwin[2:Slmax+1]**2)*(bl[2:Slmax+1]**2) `````` Matthieu Tristram committed Jan 14, 2019 484 485 486 487 488 489 `````` clthn = clth[:, 2: Slmax+1] S = np.zeros((nsto*npi, nsto*npi)) start = timeit.default_timer() for i in np.arange(npi): if timing: `````` Matthieu Tristram committed Jan 21, 2020 490 `````` progress_bar(i, npi) `````` Matthieu Tristram committed Jan 14, 2019 491 492 `````` for j in np.arange(i, npi): if temp: `````` Matthieu Tristram committed Feb 27, 2019 493 `````` pl = norm*pl0(allcosang[i, j], Slmax)[2:] `````` Matthieu Tristram committed Jan 14, 2019 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 `````` elem = np.sum((pl * clthn[0])) S[i, j] = elem S[j, i] = elem if polar: ii = i+ponpi jj = j+ponpi cij, sij = polrotangle(rpix[:, i], rpix[:, j]) cji, sji = polrotangle(rpix[:, j], rpix[:, i]) cos_chi = allcosang[i, j] # Tegmark version Q22 = norm*F1l2(cos_chi, Slmax) # # /!\ signe - ! R22 = -norm*F2l2(cos_chi, Slmax) # # Matt version # d20 = dlss(cos_chi, 2, 0, Slmax+1) # d2p2 = dlss(cos_chi, 2, 2, Slmax+1) # d2m2 = dlss(cos_chi, 2, -2, Slmax+1) # P02 = -d20[2:] # Q22 = ( d2p2 + d2m2 )[2:]/2. # R22 = ( d2p2 - d2m2 )[2:]/2. if TE or TB: P02 = -norm*F1l0(cos_chi, Slmax) elemA = 0 elemB = 0 elemC = 0 elemD = 0 if TE: elemTE = P02*clthn[3] elemA += np.sum(cji*elemTE) elemB -= np.sum(sji*elemTE) elemC += np.sum(cij*elemTE) elemD -= np.sum(sij*elemTE) if TB: elemTB = P02*clthn[5] elemA += np.sum(sji*elemTB) elemB += np.sum(cji*elemTB) elemC += np.sum(sij*elemTB) elemD += np.sum(cij*elemTB) S[i, jj] = elemA S[i, jj+npi] = elemB S[ii, j] = elemC S[ii+npi, j] = elemD S[jj, i] = elemA S[jj+npi, i] = elemB S[j, ii] = elemC S[j, ii+npi] = elemD elem1 = np.sum((cij*cji*Q22 + sij*sji*R22)*clthn[1]) elem2 = np.sum((-cij*sji*Q22 + sij*cji*R22)*clthn[1]) elem3 = np.sum((sij*sji*Q22 + cij*cji*R22)*clthn[1]) elem4 = np.sum((-sij*cji*Q22 + cij*sji*R22)*clthn[1]) elem3 += np.sum((cij*cji*Q22 + sij*sji*R22)*clthn[2]) elem4 -= np.sum((-cij*sji*Q22 + sij*cji*R22)*clthn[2]) elem1 += np.sum((sij*sji*Q22 + cij*cji*R22)*clthn[2]) elem2 -= np.sum((-sij*cji*Q22 + cij*sji*R22)*clthn[2]) if EB: elemEB = np.sum((Q22 - R22)*clthn[4]) elem1 += (cji*sij + sji*cij)*elemEB elem2 += (-sji*sij + cji*cij)*elemEB elem3 += (-sji*cij - cji*sij)*elemEB elem4 += (cji*cij - sji*sij)*elemEB S[ii, jj] = elem1 S[ii, jj+npi] = elem2 S[ii+npi, jj+npi] = elem3 S[ii+npi, jj] = elem4 S[jj, ii] = elem1 S[jj+npi, ii] = elem2 S[jj+npi, ii+npi] = elem3 S[jj, ii+npi] = elem4 return S def covth_bins_MC( ellbins, nside, ipok, allcosang, bl, clth, Slmax, nsimu, `````` Matthieu Tristram committed Jan 21, 2020 581 `````` spec, pixwin=False, timing=False): `````` Matthieu Tristram committed Jan 14, 2019 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 `````` """ Can be particularly slow on sl7. To be enhanced and extended to TT and correlations. Parameters ---------- ellbins : array of floats Lowers bounds of bins. Example : taking ellbins = (2, 3, 5) will compute the spectra for bins = (2, 3.5). nside : int Healpix map resolution ipok : array of ints Healpy pixels numbers considered bl : 1D array of floats Beam window function clth : 4D or 6D array of float Fiducial power spectra Slmax : int Maximum lmax computed for the pixel covariance pixel matrix spec : 1D array of string Spectra list pixwin : bool If True, multiplies the beam window function by the pixel window function. Default: True timing : bool If True, displays timmer. Default: False Returns ---------- Pl : ndarray of floats Normalize Legendre polynomials dS/dCl S : 2D square matrix array of floats Pixel signal covariance matrix S Example ---------- >>> import pylab >>> pylab.seed(0) >>> Pl, S = covth_bins_MC(np.array([2,3,4]), 4, ipok=np.array([0,1,3]), ... allcosang=np.linspace(0,1,13), bl=np.arange(13), ... clth=np.arange(4*13).reshape(4,-1), Slmax=11, nsimu=100, `````` Matthieu Tristram committed Jan 21, 2020 624 `````` ... spec, pixwin=True, timing=False) `````` Matthieu Tristram committed Jan 14, 2019 625 626 627 628 629 630 631 632 `````` >>> print(round(np.sum(Pl),5), round(np.sum(S),5)) (12.68135, 406990.12056) >>> import pylab >>> pylab.seed(0) >>> Pl, S = covth_bins_MC(np.array([2,3,4]), 4, ipok=np.array([0,1,3]), ... allcosang=np.linspace(0,1,13), bl=np.arange(13), ... clth=np.arange(4*13).reshape(4,-1), Slmax=11, nsimu=100, `````` Matthieu Tristram committed Jan 21, 2020 633 `````` ... spec, pixwin=True, timing=False) `````` Matthieu Tristram committed Jan 14, 2019 634 635 636 `````` >>> print(round(np.sum(Pl),5), round(np.sum(S),5)) (42.35592, 7414.01784) """ `````` Matthieu Tristram committed Jan 21, 2020 637 638 `````` polar = 'EE' in spec or 'BB' in spec `````` Matthieu Tristram committed Jan 14, 2019 639 640 641 642 643 644 645 646 647 648 `````` if nsimu == 1: nsimu = (12 * nside**2) * 10 * (int(polar) + 1) lmax = ellbins[-1] ell = np.arange(np.min(ellbins), np.max(ellbins) + 1) nbins = len(ellbins) - 1 minell = np.array(ellbins[0: nbins]) maxell = np.array(ellbins[1: nbins + 1]) - 1 ellval = (minell + maxell) * 0.5 `````` Matthieu Tristram committed Jan 21, 2020 649 `````` Stokes, spec, istokes, ispecs = getstokes(spec) `````` Matthieu Tristram committed Jan 14, 2019 650 651 `````` nspec = len(spec) ll = np.arange(Slmax+1) `````` Matthieu Tristram committed Feb 27, 2019 652 `````` fpixwin = extrapolpixwin(nside, Slmax, pixwin) `````` Matthieu Tristram committed Jan 14, 2019 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 `````` masks = [] for i in np.arange(nbins): masks.append((ll[:] >= minell[i]) & (ll[:] <= maxell[i])) masks = np.array(masks) npix = len(ipok) start = timeit.default_timer() norm = bl[0: Slmax + 1]**2 * fpixwin[0: Slmax + 1]**2 if polar: ClthOne = np.zeros((nspec * (nbins), 6, (Slmax + 1))) for l in np.arange(2 * nbins): ClthOne[l, int(l / nbins + 1)] = masks[l % nbins] * norm if corr: print("not implemented") # break; for l in np.arange(2 * nbins, 3 * nbins): ClthOne[l, 1] = masks[l % nbins] * norm ClthOne[l, 2] = masks[l % nbins] * norm ClthOne[l, 4] = masks[l % nbins] * norm Pl = np.zeros((nspec * (nbins), 2 * npix, 2 * npix)) start = timeit.default_timer() for l in np.arange((nspec * nbins)): if timing: `````` Matthieu Tristram committed Jan 21, 2020 677 `````` progress_bar(l, nspec * (nbins)) `````` Matthieu Tristram committed Jan 14, 2019 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 `````` data = [ np.array( hp.synfast( ClthOne[l], nside, lmax=Slmax, new=True, verbose=False) )[1: 3, ipok].flatten() for s in np.arange(nsimu)] Pl[l] = np.cov( np.array(data).reshape(nsimu, 2 * npix), rowvar=False) Pl = Pl.reshape(nspec, nbins, 2 * npix, 2 * npix) S = np.cov( np.array([ np.array( hp.synfast( clth[:, : Slmax + 1] * norm, nside, lmax=Slmax, new=True, verbose=False) )[1:3, ipok].flatten() for s in np.arange(nsimu)]).reshape( nsimu, 2*npix), rowvar=False) else: ClthOne = np.zeros((nbins, (Slmax + 1))) for l in np.arange((nbins)): ClthOne[l] = masks[l] * norm Pl = np.zeros(((nbins), npix, npix)) for l in np.arange((nbins)): if timing: `````` Matthieu Tristram committed Jan 21, 2020 708 `````` progress_bar(l, nspec * (nbins)) `````` Matthieu Tristram committed Jan 14, 2019 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 `````` Pl[l] = np.cov( np.array([ hp.synfast( ClthOne[l], nside, lmax=Slmax, verbose=False )[ipok] for s in np.arange(nsimu)]).reshape( nsimu, npix), rowvar=False) Pl = Pl.reshape(1, nbins, npix, npix) S = np.cov( np.array([ np.array( hp.synfast( clth[:, : Slmax + 1] * norm, nside, lmax=Slmax, new=True, verbose=False) )[0, ipok].flatten() for s in np.arange(nsimu)]).reshape( nsimu, npix), rowvar=False) stop = timeit.default_timer() return (Pl, S) def S_bins_MC( ellbins, nside, ipok, allcosang, bl, clth, Slmax, nsimu, `````` Matthieu Tristram committed Jan 21, 2020 738 `````` spec, pixwin=False, timing=False): `````` Matthieu Tristram committed Jan 14, 2019 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 `````` """ Can be particularly slow on sl7 ! To be enhanced and extended to TT and correlations Parameters ---------- ellbins : array of floats Lowers bounds of bins. Example : taking ellbins = (2, 3, 5) will compute the spectra for bins = (2, 3.5). nside : int Healpix map resolution ipok : array of ints Healpy pixels numbers considered bl : 1D array of floats Beam window function clth : 4D or 6D array of float Fiducial power spectra Slmax : int Maximum lmax computed for the pixel covariance pixel matrix polar : bool If True, get Stokes parameters for polar. Default: True temp : bool If True, get Stokes parameters for temperature. Default: False corr : bool If True, get Stokes parameters for EB and TB. Default: False pixwin : bool If True, multiplies the beam window function by the pixel window function. Default: True timing : bool If True, displays timmer. Default: False Returns ---------- S : 2D array of floats Pixel signal covariance matrix S """ `````` Matthieu Tristram committed Jan 21, 2020 776 `````` polar = 'EE' in spec or 'BB' in spec `````` Matthieu Tristram committed Jan 14, 2019 777 778 779 780 781 782 783 784 `````` if nsimu == 1: nsimu = (12 * nside**2) * 10 * (int(polar) + 1) lmax = ellbins[-1] ell = np.arange(np.min(ellbins), np.max(ellbins) + 1) nbins = len(ellbins) - 1 minell = np.array(ellbins[0: nbins]) maxell = np.array(ellbins[1: nbins + 1]) - 1 ellval = (minell + maxell) * 0.5 `````` Matthieu Tristram committed Jan 21, 2020 785 `````` Stokes, spec, ind = getstokes(spec) `````` Matthieu Tristram committed Jan 14, 2019 786 787 `````` nspec = len(spec) ll = np.arange(Slmax + 2) `````` Matthieu Tristram committed Feb 27, 2019 788 `````` fpixwin = extrapolpixwin(nside, Slmax, pixwin) `````` Matthieu Tristram committed Jan 14, 2019 789 790 791 792 793 794 `````` masks = [] for i in np.arange(nbins): masks.append((ll[:] >= minell[i]) & (ll[:] <= maxell[i])) masks = np.array(masks) npix = len(ipok) start = timeit.default_timer() `````` Matthieu Tristram committed Feb 27, 2019 795 `````` norm = bl[0: Slmax + 1]**2 * fpixwin[0: Slmax + 1]**2 `````` Matthieu Tristram committed Jan 14, 2019 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 `````` if polar: ClthOne = np.zeros((nspec * (nbins), 6, (Slmax + 2))) for l in np.arange(2 * nbins): ClthOne[l, l / nbins + 1] = masks[l % nbins] * norm if corr: print("not implemented") # break; for l in np.arange(2*nbins, 3*nbins): ClthOne[l, 1] = masks[l % nbins]*norm ClthOne[l, 2] = masks[l % nbins]*norm # couille ici : -nbins*(l/nbins)]*norm ClthOne[l, 4] = masks[l % nbins]*norm S = np.cov( np.array( [np.array(hp.synfast( clth[:, : Slmax + 2]*norm, nside, lmax=Slmax, new=True, verbose=False))[1:3, ipok].flatten() for s in np.arange(nsimu)]).reshape(nsimu, 2 * npix), rowvar=False) else: ClthOne = np.zeros((nbins, (Slmax+2))) for l in np.arange((nbins)): ClthOne[l] = masks[l]*norm S = np.cov( np.array( [np.array(hp.synfast( clth[:, : Slmax + 2] * norm, nside, lmax=Slmax, new=True, verbose=False))[0, ipok].flatten() for s in np.arange(nsimu)]).reshape(nsimu, npix), rowvar=False) return S def polrotangle(ri, rj): """ Computes cosine and sine of twice the angle between pixels i and j. Parameters ---------- ri : 3D array of floats Coordinates of vector corresponding to input pixels i following healpy.pix2vec(nside,ipix) output rj : 3D array of floats Coordinates of vector corresponding to input pixels j following healpy.pix2vec(nside,jpix) output Returns ---------- cos2a : 1D array of floats Cosine of twice the angle between pixels i and j sin2a : 1D array of floats Sine of twice the angle between pixels i and j Example ---------- >>> cos2a, sin2a = polrotangle([0.1,0.2,0.3], [0.4,0.5,0.6]) >>> print(round(cos2a,5),round(sin2a,5)) (0.06667, 0.37333) """ z = np.array([0.0, 0.0, 1.0]) # Compute ri^rj : unit vector for the great circle connecting i and j rij = np.cross(ri, rj) norm = np.sqrt(np.dot(rij, np.transpose(rij))) # case where pixels are identical or diametrically opposed on the sky if norm <= 1e-15: cos2a = 1.0 sin2a = 0.0 return cos2a, sin2a rij = rij / norm # Compute z^ri : unit vector for the meridian passing through pixel i ris = np.cross(z, ri) norm = np.sqrt(np.dot(ris, np.transpose(ris))) # case where pixels is at the pole if norm <= 1e-15: cos2a = 1.0 sin2a = 0.0 return cos2a, sin2a ris = ris / norm # Now, the angle we want is that # between these two great circles: defined by cosa = np.dot(rij, np.transpose(ris)) # the sign is more subtle : see tegmark et de oliveira costa 2000 eq. A6 rijris = np.cross(rij, ris) sina = np.dot(rijris, np.transpose(ri)) # so now we have directly cos2a and sin2a cos2a = 2.0 * cosa * cosa - 1.0 sin2a = 2.0 * cosa * sina return cos2a, sin2a def dlss(z, s1, s2, lmax): """ Computes the reduced Wigner D-function d^l_ss' Parameters ---------- z : float Cosine of the angle between two pixels s1 : int Spin number 1 s2 : int Spin number 2 lmax : int Maximum multipole Returns ---------- d : 1D array of floats ??? Example ---------- >>> d = dlss(0.1, 2, 2, 5) >>> print(round(sum(d),5)) 0.24351 """ d = np.zeros((lmax + 1)) if s1 < abs(s2): print("error spins, s1<|s2|") return # Conv: sign = -1 if (s1 + s2) and 1 else 1 sign = (-1)**(s1 - s2) fs1 = math.factorial(2.0 * s1) fs1ps2 = math.factorial(1.0 * s1 + s2) fs1ms2 = math.factorial(1.0 * s1 - s2) num = (1.0 + z)**(0.5 * (s1 + s2)) * (1.0 - z)**(0.5 * (s1 - s2)) # Initialise the recursion (l = s1 + 1) d[s1] = sign / 2.0**s1 * np.sqrt(fs1 / fs1ps2 / fs1ms2) * num l1 = s1 + 1.0 rhoSSL1 = np.sqrt((l1 * l1 - s1 * s1) * (l1 * l1 - s2 * s2)) / l1 d[s1+1] = (2 * s1 + 1.0)*(z - s2 / (s1 + 1.0)) * d[s1] / rhoSSL1 # Build the recursion for l > s1 + 1 for l in np.arange(s1 + 1, lmax, 1): l1 = l + 1.0 numSSL = (l * l * 1.0 - s1 * s1) * (l * l * 1.0 - s2 * s2) rhoSSL = np.sqrt(numSSL) / (l * 1.0) numSSL1 = (l1 * l1 - s1 * s1) * (l1 * l1 - s2 * s2) rhoSSL1 = np.sqrt(numSSL1) / l1 numd = (2.0 * l + 1.0) * (z - s1 * s2 / (l * 1.0) / l1) * d[l] d[l+1] = (numd - rhoSSL * d[l-1]) / rhoSSL1 return d def pl0(z, lmax): """ Computes the associated Legendre function of the first kind of order 0 Pn(z) from 0 to lmax (inclusive). Parameters ---------- z : float Cosine of the angle between two pixels lmax : int Maximum multipole Returns ---------- Pn : 1D array of floats Legendre function Example ---------- >>> thepl0 = pl0(0.1, 5) >>> print(round(sum(thepl0),5)) 0.98427 """ Pn = special.lpn(lmax, z)[0] return Pn def pl2(z, lmax): """ Computes the associated Legendre function of the first kind of order 2 from 0 to lmax (inclusive) Parameters ---------- z : float Cosine of the angle between two pixels lmax : int Maximum multipole Returns ---------- Pn2 : 1D array of floats Legendre function Example ---------- >>> thepl2 = pl2(0.1, 5) >>> print(round(sum(thepl2),5)) -7.49183 """ Pn2 = special.lpmn(2, lmax, z)[0][2] return Pn2 # ####### F1 and F2 functions from Tegmark & De Oliveira-Costa, 2000 ######### def F1l0(z, lmax): """ Compute the F1l0 function from Tegmark & De Oliveira-Costa, 2000 Parameters ---------- z : float Cosine of the angle between two pixels lmax : int Maximum multipole Returns ---------- bla : 1D array of float F1l0 function from Tegmark & De Oliveira-Costa, 2000 Example ---------- >>> theF1l0= F1l0(0.1, 5) >>> print(round(sum(theF1l0),5)) 0.20392 """ if abs(z) == 1.0: return(np.zeros(lmax - 1)) else: ell = np.arange(2, lmax + 1) thepl = pl0(z, lmax) theplm1 = np.append(0, thepl[:-1]) thepl = thepl[2:] theplm1 = theplm1[2:] a0 = 2.0 / np.sqrt((ell - 1) * ell * (ell + 1) * (ell + 2)) a1 = ell * z * theplm1 / (1 - z**2) a2 = (ell / (1 - z**2) + ell * (ell - 1) / 2) * thepl bla = a0 * (a1 - a2) return bla def F1l2(z, lmax): """ Compute the F1l2 function from Tegmark & De Oliveira-Costa, 2000 Parameters ---------- z : float Cosine of the angle between two pixels lmax : int Maximum multipole Returns ---------- bla : 1D array of float F1l2 function from Tegmark & De Oliveira-Costa, 2000 Example ---------- >>> theF1l2= F1l2(0.1, 5) >>> print(round(sum(theF1l2),5)) 0.58396 """ if z == 1.0: return np.ones(lmax - 1) * 0.5 elif z == -1.0: ell = np.arange(lmax + 1) return 0.5 * (-1)**ell[2:] else: ell = np.arange(2, lmax + 1) thepl2 = pl2(z, lmax) theplm1_2 = np.append(0, thepl2[:-1]) thepl2 = thepl2[2:] theplm1_2 = theplm1_2[2:] a0 = 2.0 / ((ell - 1) * ell * (ell + 1) * (ell + 2)) a1 = (ell + 2) * z * theplm1_2 / (1 - z**2) a2 = ((ell - 4) / (1 - z**2) + ell * (ell - 1) / 2) * thepl2 bla = a0 * (a1 - a2) return bla def F2l2(z, lmax): """ Compute the F2l2 function from Tegmark & De Oliveira-Costa, 2000 Parameters ---------- z : float Cosine of the angle between two pixels lmax : int Maximum multipole Returns ---------- ??? Example ---------- >>> theF2l2= F2l2(0.1, 5) >>> print(round(sum(theF2l2),5)) 0.34045 """ if z == 1.0: return -0.5 * np.ones(lmax - 1) elif z == -1.0: ell = np.arange(lmax + 1) return 0.5 * (-1)**ell[2:] else: ell = np.arange(2, lmax + 1) thepl2 = pl2(z, lmax) theplm1_2 = np.append(0, thepl2[:-1]) thepl2 = thepl2[2:] theplm1_2 = theplm1_2[2:] a0 = 4.0 / ((ell - 1) * ell * (ell + 1) * (ell + 2) * (1 - z**2)) a1 = (ell + 2) * theplm1_2 a2 = (ell - 1) * z * thepl2 bla = a0 * (a1 - a2) return bla `````` Matthieu Tristram committed Jan 14, 2019 1132 1133 `````` `````` Matthieu Tristram committed Jan 14, 2019 1134 1135 1136 1137 ``````if __name__ == "__main__": """ Run the doctest using `````` Matthieu Tristram committed Feb 27, 2019 1138 `````` python libcov.py' `````` Matthieu Tristram committed Jan 14, 2019 1139 1140 1141 1142 1143 1144 1145 1146 `````` If the tests are OK, the script should exit gracefuly, otherwise the failure(s) will be printed out. """ import doctest if np.__version__ >= "1.14.0": np.set_printoptions(legacy="1.13") doctest.testmod()``````