# WikiStart: statdocs.txt

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1 | |

2 | ----- Documentation for Stat\cov2cor.pro ----- |

3 | NAME: |

4 | COV2COR |

5 | |

6 | PURPOSE: |

7 | CONVERT COVARIANCE MATRIX TO MATRIX OF CORRELATION COEFFICIENTS |

8 | |

9 | CATEGORY: |

10 | statistics |

11 | |

12 | CALLING SEQUENCE: |

13 | COV2COR, COV, SIGMA, COR |

14 | |

15 | INPUTS: |

16 | COV = COVARIANCE MATRIX |

17 | |

18 | OUTPUTS: |

19 | SIGMA = ERROR ARRAY |

20 | COR = CORRELATION MATRIX |

21 | |

22 | COMMON BLOCKS: |

23 | NONE |

24 | |

25 | SIDE EFFECTS: |

26 | NONE |

27 | |

28 | REQUIREMENTS: |

29 | COR MUST BE SQUARE, SYMMETRIC, WITH POSITIVE DIAGONALS |

30 | |

31 | PROCEDURE: |

32 | COV = COR / (SIGMA # SIGMA) |

33 | WITH: SIGMA = SQRT(DIAG(COV)) |

34 | |

35 | LOCAL PROCEDURE CALLED: |

36 | NONE |

37 | |

38 | LOCAL FUNCTION USED: |

39 | NONE |

40 | |

41 | LOCAL SYSTEM VARIABLE USED: |

42 | NONE |

43 | |

44 | REVISION HISTORY: |

45 | Written by pcr 2010/02/01 |

46 | from www.boulder.swri.edu/~layoung/idl/layoung_v5/math/cov2cor.pro |

47 | |

48 | ----- Documentation for Stat\get_outlier_fences.pro ----- |

49 | NAME: |

50 | GET_OUTLIER_FENCES |

51 | |

52 | AUTHOR: |

53 | pierre.cruzalebes@oca.eu |

54 | |

55 | PURPOSE: |

56 | return lower and upper fences identifying extreme values in tails |

57 | of univariate distribution (outliers). |

58 | Estimation of percentiles following NIST-recommended method |

59 | http://www.itl.nist.gov/div898/handbook/prc/section2/prc252.htm |

60 | For small nber of data, estimate percentiles using normal CDF |

61 | |

62 | CATEGORY: |

63 | statistics |

64 | |

65 | CALLING SEQUENCE: |

66 | FENCES=GET_OUTLIER_FENCES(X,ALFA,COEFF) |

67 | |

68 | INPUTS: |

69 | X = experimental data distribution. |

70 | ALFA = percentage of experimental data included in the interpercentile range. |

71 | (must be gt 0 and lt 1, e.g. =0.5 for quartile-based outliers). |

72 | COEFF = outlier coefficient defining types of outliers |

73 | (=1.5 for mild outliers, =3 for extreme outliers). |

74 | |

75 | OUTPUTS: |

76 | Function result = FENCES = 2-element array of outlier fences |

77 | (FENCES(0)=lower fence, FENCES(1)=upper fence). |

78 | |

79 | OPTIONAL OUTPUT PARAMETERS: |

80 | NONE. |

81 | |

82 | COMMON BLOCKS: |

83 | NONE. |

84 | |

85 | SIDE EFFECTS: |

86 | NONE. |

87 | |

88 | RESTRICTIONS: |

89 | NONE. |

90 | |

91 | PROCEDURE: |

92 | |

93 | LOCAL PROCEDURE CALLED: |

94 | NONE. |

95 | |

96 | LOCAL FUNCTION USED: |

97 | NONE. |

98 | |

99 | LOCAL SYSTEM VARIABLE USED: |

100 | NB_DATA_BOOT |

101 | |

102 | REVISION HISTORY: |

103 | Written by pcr 2009/02/02 |

104 | Last modification by pcr 2010/12/12 |

105 | |

106 | ----- Documentation for Stat\loca_scale_chi2.pro ----- |

107 | NAME: |

108 | LOCA_SCALE_CHI2 |

109 | |

110 | AUTHOR: |

111 | pierre.cruzalebes@oca.eu |

112 | |

113 | PURPOSE: |

114 | return location and scale parameters of standard chi-square distribution |

115 | for experimental data, using median (for location) and interpercentile range (for scale). |

116 | interquartile range (ALFA=0.75) usually used as robust measure of scale |

117 | Estimation of percentiles following NIST-recommended method |

118 | http://www.itl.nist.gov/div898/handbook/prc/section2/prc252.htm |

119 | |

120 | CATEGORY: |

121 | statistics |

122 | |

123 | CALLING SEQUENCE: |

124 | LOCA_SCALE_CHI2,X,ALFA,NU,LOC,SCA |

125 | |

126 | INPUTS: |

127 | X = experimental data set. |

128 | ALFA = percentage of experimental data included in the interpercentile range (must be gt 0 and lt 1). |

129 | NU = number of degrees of freedom. |

130 | |

131 | OUTPUTS: |

132 | LOC = location parameter of standard chi-square distribution. |

133 | SCA = scale parameter of standard chi-square distribution. |

134 | |

135 | OPTIONAL OUTPUT PARAMETERS: |

136 | NONE. |

137 | |

138 | COMMON BLOCKS: |

139 | NONE. |

140 | |

141 | SIDE EFFECTS: |

142 | NONE. |

143 | |

144 | RESTRICTIONS: |

145 | NONE. |

146 | |

147 | PROCEDURE: |

148 | |

149 | LOCAL PROCEDURE CALLED: |

150 | NONE |

151 | |

152 | LOCAL FUNCTION USED: |

153 | NONE |

154 | |

155 | LOCAL SYSTEM VARIABLE USED: |

156 | NONE |

157 | |

158 | REVISION HISTORY: |

159 | Written by pcr 2008/01/24 |

160 | last modification by pcr 2009/09/31 |

161 | |

162 | ----- Documentation for Stat\loca_scale_norm.pro ----- |

163 | NAME: |

164 | LOCA_SCALE_NORM |

165 | |

166 | AUTHOR: |

167 | pierre.cruzalebes@oca.eu |

168 | |

169 | PURPOSE: |

170 | return location and scale parameters of standard normal distribution |

171 | for experimental data, using median (for location) and interpercentile |

172 | range (for scale). |

173 | Estimation of percentiles following NIST-recommended method |

174 | http://www.itl.nist.gov/div898/handbook/prc/section2/prc252.htm |

175 | |

176 | CATEGORY: |

177 | statistics |

178 | |

179 | CALLING SEQUENCE: |

180 | LOCA_SCALE_NORM,X,ALFA,LOC,SCA |

181 | |

182 | INPUTS: |

183 | X = experimental data set. |

184 | ALFA = percentage of experimental data included in the interpercentile |

185 | range (must be gt 0 and lt 1). |

186 | |

187 | OUTPUT: |

188 | LOC = location parameter of standard normal distribution. |

189 | SCA = scale parameter of standard normal distribution. |

190 | |

191 | OPTIONAL OUTPUT PARAMETERS: |

192 | NONE. |

193 | |

194 | COMMON BLOCKS: |

195 | NONE. |

196 | |

197 | SIDE EFFECTS: |

198 | NONE. |

199 | |

200 | RESTRICTIONS: |

201 | NONE. |

202 | |

203 | PROCEDURE: |

204 | |

205 | LOCAL PROCEDURE CALLED: |

206 | NONE |

207 | |

208 | LOCAL FUNCTION USED: |

209 | NONE |

210 | |

211 | LOCAL SYSTEM VARIABLE USED: |

212 | NONE |

213 | |

214 | REVISION HISTORY: |

215 | Written by pcr 2008/05/31 |

216 | last modification by pcr 2009/04/14 |

217 | |

218 | ----- Documentation for Stat\mem_stdev.pro ----- |

219 | NAME: |

220 | MEM_STDEV |

221 | |

222 | AUTHOR: |

223 | pierre.cruzalebes@oca.eu |

224 | |

225 | PURPOSE: |

226 | compute standard deviation of asymmetric distribution from maximum entropy principle. |

227 | |

228 | CATEGORY: |

229 | statistics |

230 | |

231 | CALLING SEQUENCE: |

232 | MEMSTD = MEM_STDEV(LOWER_ERROR,UPPER_ERROR) |

233 | |

234 | INPUTS: |

235 | LOWER_ERROR = lower error. |

236 | UPPER_ERROR = upper error. |

237 | |

238 | OPTIONAL INPUT PARAMETER: |

239 | none. |

240 | |

241 | OUTPUTS: |

242 | Function result = MEMSTD = standard deviation of distribution. |

243 | |

244 | OPTIONAL OUTPUT PARAMETER: |

245 | NONE. |

246 | |

247 | COMMON BLOCKS: |

248 | NONE. |

249 | |

250 | SIDE EFFECTS: |

251 | NONE. |

252 | |

253 | RESTRICTIONS: |

254 | NONE. |

255 | |

256 | PROCEDURE: |

257 | memvar=MEMSTD^2=UPPER_ERROR*LOWER_ERROR-(UPPER_ERROR-LOWER_ERROR)/lambda |

258 | where lambda is computed according to JCGM 100:2008, section 4.3.8, note 2 |

259 | |

260 | LOCAL PROCEDURE CALLED: |

261 | NONE |

262 | |

263 | LOCAL FUNCTION USED: |

264 | NONE |

265 | |

266 | LOCAL SYSTEM VARIABLE USED: |

267 | NONE |

268 | |

269 | REVISION HISTORY: |

270 | Written by pcr 2009/10/05 |

271 | Last modification by pcr 2010/12/07 |

272 | |

273 | ----- Documentation for Stat\plot_distri.pro ----- |

274 | NAME: |

275 | PLOT_DISTRI |

276 | |

277 | AUTHOR: |

278 | pierre.cruzalebes@oca.eu |

279 | |

280 | PURPOSE: |

281 | plot chi-square and parameter statistics, used with fit_model and fit_sed |

282 | |

283 | CATEGORY: |

284 | statistics |

285 | |

286 | CALLING SEQUENCE: |

287 | PLOT_DISTRI,DELTACHI2,NU,PARA,FTIT,OUTPUT_FILE_STEM,WSKIP,CLEVEL |

288 | |

289 | INPUTS: |

290 | DELTACHI2 = row vector of reduced chi-square differences. |

291 | NU = number of degrees of freedom. |

292 | FTIT = row vector of string chains for plot of free parameter axis titles. |

293 | PARA = 2-dim array of free model parameters, sorted in ascending order |

294 | of chi-square diff, length=length(DELTACHI2)*length(FTIT). |

295 | OUTPUT_FILE_STEM = output file stem for writing in postscript files. |

296 | WSKIP = index of first newly created window in current procedure. |

297 | CLEVEL = confidence level. |

298 | |

299 | OUTPUT: |

300 | WSKIP = index of next newly created window after current procedure. |

301 | |

302 | OPTIONAL OUTPUT PARAMETERS: |

303 | NONE. |

304 | |

305 | COMMON BLOCKS: |

306 | NONE. |

307 | |

308 | SIDE EFFECTS: |

309 | NONE. |

310 | |

311 | RESTRICTIONS: |

312 | NONE. |

313 | |

314 | PROCEDURE: |

315 | uniform order statistic medians (quantiles) given by |

316 | http://www.itl.nist.gov/div898/handbook/eda/section3/probplot.htm |

317 | standardization of mean-squares from |

318 | Wilson & Hilferty, Proc. of Nat. Acad. of Sciences of USA, 17, 684-688 (1931) |

319 | |

320 | LOCAL PROCEDURE CALLED: |

321 | LOCA_SCALE_CHI2 |

322 | RESET_PLOT |

323 | |

324 | LOCAL FUNCTION USED: |

325 | NONE |

326 | |

327 | LOCAL SYSTEM VARIABLE USED: |

328 | OBS_PATH |

329 | PLOT_DEV |

330 | |

331 | REVISION HISTORY: |

332 | Written by pcr 2008/04/08 |

333 | last modification by pcr 2011/01/14 |

334 | |

335 | ----- Documentation for Stat\rect_stdev.pro ----- |

336 | NAME: |

337 | RECT_STDEV |

338 | |

339 | AUTHOR: |

340 | pierre.cruzalebes@oca.eu |

341 | |

342 | PURPOSE: |

343 | compute standard deviation of rectangular (uniform) distribution. |

344 | |

345 | CATEGORY: |

346 | statistics |

347 | |

348 | CALLING SEQUENCE: |

349 | RECSTD = RECT_STDEV(LOWER_ERROR,UPPER_ERROR,CONFID_LEVEL,LOWER_BOUND,UPPER_BOUND) |

350 | |

351 | INPUTS: |

352 | LOWER_ERROR = lower error. |

353 | UPPER_ERROR = upper error. |

354 | CONFID_LEVEL = level of confidence. |

355 | |

356 | OPTIONAL INPUT PARAMETER: |

357 | none. |

358 | |

359 | OUTPUTS: |

360 | Function result = RECSTD = standard deviation of distribution. |

361 | |

362 | OPTIONAL OUTPUT PARAMETER: |

363 | NONE. |

364 | |

365 | COMMON BLOCKS: |

366 | NONE. |

367 | |

368 | SIDE EFFECTS: |

369 | NONE. |

370 | |

371 | RESTRICTIONS: |

372 | NONE. |

373 | |

374 | PROCEDURE: |

375 | recvar=RECSTD^2=cover^2*(UPPER_ERROR+LOWER_ERROR)^2/12 |

376 | where cover=sqrt(2)/3*inv_erf(CONFID_LEVEL) |

377 | |

378 | LOCAL PROCEDURE CALLED: |

379 | NONE |

380 | |

381 | LOCAL FUNCTION USED: |

382 | NONE |

383 | |

384 | LOCAL SYSTEM VARIABLE USED: |

385 | NONE |

386 | |

387 | REVISION HISTORY: |

388 | Written by pcr 2009/10/15 |

389 | |

390 | ----- Documentation for Stat\tri_stdev.pro ----- |

391 | NAME: |

392 | TRI_STDEV |

393 | |

394 | AUTHOR: |

395 | pierre.cruzalebes@oca.eu |

396 | |

397 | PURPOSE: |

398 | compute standard deviation and upper/lower bounds of triangular |

399 | distribution from quantiles. |

400 | |

401 | CATEGORY: |

402 | statistics |

403 | |

404 | CALLING SEQUENCE: |

405 | TRISTD = TRI_STDEV(MODE_VAL,LOWER_QUANT,UPPER_QUANT,LOWER_PROB, |

406 | UPPER_PROB,UPPER_BOUND,LOWER_BOUND) |

407 | |

408 | INPUTS: |

409 | MODE_VAL = mode value. |

410 | LOWER_QUANT = lower quantile. |

411 | UPPER_QUANT = upper quantile. |

412 | LOWER_PROB = probability of lower quantile. |

413 | UPPER_QUANT = probability of upper quantile. |

414 | |

415 | OPTIONAL INPUT PARAMETER: |

416 | none. |

417 | |

418 | OUTPUTS: |

419 | Function result = TRISTD = standard deviation of |

420 | triangular distribution. |

421 | UPPER_BOUND = upper limit of triangular distribution |

422 | LOWER_BOUND = lower limit |

423 | |

424 | OPTIONAL OUTPUT PARAMETER: |

425 | NONE. |

426 | |

427 | COMMON BLOCKS: |

428 | NONE. |

429 | |

430 | SIDE EFFECTS: |

431 | NONE. |

432 | |

433 | RESTRICTIONS: |

434 | NONE. |

435 | |

436 | PROCEDURE: |

437 | trivar=TRISTD^2=(LOWER_BOUND^2+MODE_VAL^2+UPPER_BOUND^2 |

438 | -LOWER_BOUND*MODE_VAL-UPPER_BOUND*MODE_VAL |

439 | -LOWER_BOUND*UPPER_BOUND)/18 |

440 | where LOWER_BOUND and UPPER_BOUND are triangular distribution |

441 | limits calculated from upper and lower quantiles |

442 | according to Kotz & Van Dorp (2004) section 1.6 |

443 | |

444 | LOCAL PROCEDURE CALLED: |

445 | NONE |

446 | |

447 | LOCAL FUNCTION USED: |

448 | NONE |

449 | |

450 | LOCAL SYSTEM VARIABLE USED: |

451 | EPSILON |

452 | |

453 | REVISION HISTORY: |

454 | Written by pcr 2009/10/05 |

455 | last modification by pcr 2010/12/07 |

456 | |

457 | ----- Documentation for Stat\wavg.pro ----- |

458 | NAME: |

459 | WAVG |

460 | |

461 | AUTHOR: |

462 | pierre.cruzalebes@oca.eu |

463 | |

464 | PURPOSE: |

465 | compute weighted mean of data vector. |

466 | |

467 | CATEGORY: |

468 | statistics |

469 | |

470 | CALLING SEQUENCE: |

471 | WMEAN = WAVG(WEIGHT,DATA) |

472 | |

473 | INPUTS: |

474 | WEIGHT = row vector of weights . |

475 | DATA = row vector of data (same length as WEIGHT). |

476 | |

477 | OUTPUTS: |

478 | Function result = WMEAN = weighted mean. |

479 | |

480 | OPTIONAL OUTPUT PARAMETER: |

481 | NONE. |

482 | |

483 | COMMON BLOCKS: |

484 | NONE. |

485 | |

486 | SIDE EFFECTS: |

487 | NONE. |

488 | |

489 | RESTRICTIONS: |

490 | NONE. |

491 | |

492 | PROCEDURE: |

493 | |

494 | LOCAL PROCEDURE CALLED: |

495 | NONE |

496 | |

497 | LOCAL FUNCTION USED: |

498 | NONE |

499 | |

500 | LOCAL SYSTEM VARIABLE USED: |

501 | NONE |

502 | |

503 | REVISION HISTORY: |

504 | Written by pcr 2008/05/27 |

505 | last modification by pcr 2010/11/10 |

506 | |

507 | ----- Documentation for Stat\wstdev.pro ----- |

508 | NAME: |

509 | WSTDEV |

510 | |

511 | AUTHOR: |

512 | pierre.cruzalebes@oca.eu |

513 | |

514 | PURPOSE: |

515 | compute weighted mean of data vector |

516 | and weighted standard deviation (or mean standard weighted error). |

517 | |

518 | CATEGORY: |

519 | statistics |

520 | |

521 | CALLING SEQUENCE: |

522 | WSTD = WSTDEV(WEIGHT,DATA,WMEAN,ERROR) |

523 | |

524 | INPUTS: |

525 | WEIGHT = row vector of weights . |

526 | DATA = row vector of data (same length as WEIGHT). |

527 | |

528 | OPTIONAL INPUT PARAMETER: |

529 | ERROR = row vector of errors (same length as WEIGHT). |

530 | |

531 | OUTPUTS: |

532 | Function result = WSTD = unbiased estimator of weighted standard deviation (or mean weighted error). |

533 | WMEAN = weighted mean. |

534 | |

535 | OPTIONAL OUTPUT PARAMETER: |

536 | NONE. |

537 | |

538 | COMMON BLOCKS: |

539 | NONE. |

540 | |

541 | SIDE EFFECTS: |

542 | NONE. |

543 | |

544 | RESTRICTIONS: |

545 | NONE. |

546 | |

547 | PROCEDURE: |

548 | WMEAN=SUM(WEIGHT*DATA)/SUM(WEIGHT) |

549 | following http://pygsl.sourceforge.net/reference/pygsl/node52.html |

550 | and http://www.gnu.org/software/gsl/manual/html_node/Weighted-Samples.html : |

551 | WSTDEV^2=SUM(WEIGHT)/((SUM(WEIGHT))^2-SUM(WEIGHT^2)) |

552 | *SUM(WEIGHT*(DATA-WMEAN)^2) |

553 | if ERROR vector provided : |

554 | WSTDEV^2=SUM(WEIGHT^2*ERROR^2)/(SUM(WEIGHT))^2 |

555 | |

556 | LOCAL PROCEDURE CALLED: |

557 | NONE |

558 | |

559 | LOCAL FUNCTION USED: |

560 | NONE |

561 | |

562 | LOCAL SYSTEM VARIABLE USED: |

563 | NONE |

564 | |

565 | REVISION HISTORY: |

566 | Written by pcr 2008/05/27 |

567 | last modification by pcr 2010/11/13 |

568 | |

569 | ----- Documentation for Stat\wsterr.pro ----- |

570 | NAME: |

571 | WSTERR |

572 | |

573 | AUTHOR: |

574 | pierre.cruzalebes@oca.eu |

575 | |

576 | PURPOSE: |

577 | compute weighted mean of data vector |

578 | and standard error on weighted mean |

579 | |

580 | CATEGORY: |

581 | statistics |

582 | |

583 | CALLING SEQUENCE: |

584 | WSTE = WSTERR(WEIGHT,DATA,WMEAN) |

585 | |

586 | INPUTS: |

587 | WEIGHT = row vector of weights . |

588 | DATA = row vector of data (same length as WEIGHT). |

589 | |

590 | OPTIONAL INPUT PARAMETER: |

591 | NONE. |

592 | |

593 | OUTPUTS: |

594 | Function result = WSTE = unbiased estimator of standard error |

595 | on weighted mean. |

596 | WMEAN = weighted mean. |

597 | |

598 | OPTIONAL OUTPUT PARAMETER: |

599 | NONE. |

600 | |

601 | COMMON BLOCKS: |

602 | NONE. |

603 | |

604 | SIDE EFFECTS: |

605 | NONE. |

606 | |

607 | RESTRICTIONS: |

608 | NONE. |

609 | |

610 | PROCEDURE: |

611 | WMEAN=SUM(WEIGHT*DATA)/SUM(WEIGHT) |

612 | following Gatz & Smith, Atmosph. Env. 29, 11, pp. 1185-1193 (1995) : |

613 | WSTERR^2 = (N/N-1)/(SUM(WEIGHT))^2 |

614 | * ( TOTAL((WEIGHT*DATA-AVG(WEIGHT)*WMEAN)^2) |

615 | - 2.*WMEAN*TOTAL((WEIGHT-AVG(WEIGHT)) |

616 | * (WEIGHT*DATA-AVG(WEIGHT)*WMEAN)) |

617 | + WMEAN^2*TOTAL((WEIGHT-AVG(WEIGHT))^2) ) |

618 | |

619 | LOCAL PROCEDURE CALLED: |

620 | NONE |

621 | |

622 | LOCAL FUNCTION USED: |

623 | NONE |

624 | |

625 | LOCAL SYSTEM VARIABLE USED: |

626 | NONE |

627 | |

628 | MODIFICATION HISTORY: |

629 | Written by pcr 2010/11/13 |

630 |