error_utils.py 23.1 KB
Newer Older
1
#!/usr/bin/env python
Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
2 3
# Ivana Chingovska <ivana.chingovska@idiap.ch>
# Fri Dec  7 12:33:37 CET 2012
4 5 6 7
"""Utility functions for computation of EPSC curve and related measurement"""

import bob.measure
import numpy
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
8 9 10 11
from bob.measure import far_threshold, eer_threshold, min_hter_threshold, farfrr
from bob.bio.base.score.load import four_column
from collections import defaultdict
import re
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
12 13


Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
14
def calc_threshold(method, pos, negs, all_negs, far_value=None, is_sorted=False):
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
15 16 17 18 19 20 21 22
    """Calculates the threshold based on the given method.

    Parameters
    ----------
    method : str
        One of ``bpcer20``, ``eer``, ``min-hter``.
    pos : array_like
        The positive scores. They should be sorted!
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
23 24 25 26 27 28 29 30 31 32
    negs : list
        A list of array_like negative scores. Each item in the list corresponds to
        scores of one PAI.
    all_negs : array_like
        An array of all negative scores. This can be calculated from negs as well but we
        ask for it since you might have it already calculated.
    far_value : None, optional
        If method is far, far_value and all_negs are used to calculate the threshold.
    is_sorted : bool, optional
        If True, it means all scores are sorted and no sorting will happen.
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
33 34 35 36 37 38 39 40 41 42 43 44

    Returns
    -------
    float
        The calculated threshold.

    Raises
    ------
    ValueError
        If method is unknown.
    """
    method = method.lower()
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
45 46 47 48 49 50 51 52 53
    if "bpcer" in method:
        desired_apcer = 1 / float(method.replace("bpcer", ""))
        threshold = apcer_threshold(desired_apcer, pos, *negs, is_sorted=is_sorted)
    elif method == "far":
        threshold = far_threshold(all_negs, pos, far_value, is_sorted=is_sorted)
    elif method == "eer":
        threshold = eer_threshold(all_negs, pos, is_sorted=is_sorted)
    elif method == "min-hter":
        threshold = min_hter_threshold(all_negs, pos, is_sorted=is_sorted)
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
54 55 56 57
    else:
        raise ValueError("Unknown threshold criteria: {}".format(method))

    return threshold
58 59 60 61 62


def calc_pass_rate(threshold, attacks):
    """Calculates the rate of successful spoofing attacks

63 64 65 66 67 68 69 70 71 72 73 74 75
    Parameters
    ----------
    threshold :
      the threshold used for classification
    scores :
      numpy with the scores of the spoofing attacks

    Returns
    -------
    float
      rate of successful spoofing attacks
    """
    return (attacks >= threshold).mean()
76 77


Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
78
def weighted_neg_error_rate_criteria(data, weight, thres, beta=0.5, criteria="eer"):
79 80 81
    """Given the single value for the weight parameter balancing between
    impostors and spoofing attacks and a threshold, calculates the error rates
    and their relationship depending on the criteria (difference in case of
82
    'eer', hter in case of 'min-hter' criteria)
83 84
    Keyword parameters:

85 86 87 88 89
      - data - the development data used to determine the threshold. List on 4
      numpy.arrays containing: negatives (licit), positives (licit),
      negatives (spoof), positivies (spoof)
      - weight - the weight parameter balancing between impostors and spoofing
      attacks
90
      - thres - the given threshold
91 92 93
      - beta - the weight parameter balancing between real accesses and all the
      negative samples (impostors and spoofing attacks). Note that this
      parameter will be overriden and not considered if the selected criteria
94 95
      is 'min-hter'.
      - criteria - 'eer', 'wer' or 'min-hter' criteria for decision threshold
96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
  """

    licit_neg = data[0]
    licit_pos = data[1]
    spoof_neg = data[2]
    spoof_pos = data[3]  # unpacking the data
    farfrr_licit = bob.measure.farfrr(licit_neg, licit_pos, thres)
    farfrr_spoof = bob.measure.farfrr(spoof_neg, spoof_pos, thres)

    frr = farfrr_licit[1]  # farfrr_spoof[1] should have the same value
    far_i = farfrr_licit[0]
    far_s = farfrr_spoof[0]

    far_w = (1 - weight) * far_i + weight * far_s

Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
111
    if criteria == "eer":
112 113 114
        if beta == 0.5:
            return abs(far_w - frr)
        else:
Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
115
            # return abs(far_w - frr)
116 117
            return abs((1 - beta) * frr - beta * far_w)

Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
118
    elif criteria == "min-hter":
119 120 121 122 123 124
        return (far_w + frr) / 2

    else:
        return (1 - beta) * frr + beta * far_w


Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
125
def recursive_thr_search(data, span_min, span_max, weight, beta=0.5, criteria="eer"):
126 127 128 129 130
    """Recursive search for the optimal threshold given a criteria. It
    evaluates the full range of thresholds at 100 points, and computes the one
    which optimizes the threshold. In the next search iteration, it examines
    the region around the point that optimizes the threshold. The procedure
    stops when the search range is smaller then 1e-10.
131 132

  Keyword arguments:
133 134 135
    - data - the development data used to determine the threshold. List on 4
    numpy.arrays containing: negatives (licit), positives (licit), negatives
    (spoof), positivies (spoof)
136 137
    - span_min - the minimum of the search range
    - span_max - the maximum of the search range
138 139 140 141 142
    - weight - the weight parameter balancing between impostors and spoofing
    attacks
    - beta - the weight parameter balancing between real accesses and all the
    negative samples (impostors and spoofing attacks). Note that methods called
    within this function will override this parameter and not considered if the
143
    selected criteria is 'min-hter'.
144
    - criteria - the decision threshold criteria ('eer' for EER, 'wer' for
145
    Minimum WER or 'min-hter' for Minimum HTER criteria).
146 147 148 149 150 151 152 153
  """

    quit_thr = 1e-10
    steps = 100
    if abs((span_max - span_min) / span_max) < quit_thr:
        return span_max  # or span_min, it doesn't matter
    else:
        step_size = (span_max - span_min) / steps
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174
        thresholds = numpy.array([(i * step_size) + span_min for i in range(steps + 1)])
        weighted_error_rates = numpy.array(
            [
                weighted_neg_error_rate_criteria(data, weight, thr, beta, criteria)
                for thr in thresholds
            ]
        )
        selected_thres = thresholds[
            numpy.where(weighted_error_rates == min(weighted_error_rates))
        ]  # all the thresholds which have minimum weighted error rate
        thr = selected_thres[
            int(selected_thres.size / 2)
        ]  # choose the centrally positioned threshold
        return recursive_thr_search(
            data, thr - step_size, thr + step_size, weight, beta, criteria
        )


def weighted_negatives_threshold(
    licit_neg, licit_pos, spoof_neg, spoof_pos, weight, beta=0.5, criteria="eer"
):
175
    """Calculates the threshold for achieving the given criteria between the
Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
176
    FAR_w and the FRR, given the single value for the weight parameter
177 178 179
    balancing between impostors and spoofing attacks and a single value for the
    parameter beta balancing between the real accesses and the negatives
    (impostors and spoofing attacks)
180 181 182 183 184 185

  Keyword parameters:
    - licit_neg - numpy.array of scores for the negatives (licit scenario)
    - licit_pos - numpy.array of scores for the positives (licit scenario)
    - spoof_neg - numpy.array of scores for the negatives (spoof scenario)
    - spoof_pos - numpy.array of scores for the positives (spoof scenario)
186 187 188 189 190
    - weight - the weight parameter balancing between impostors and spoofing
    attacks
    - beta - the weight parameter balancing between real accesses and all the
    negative samples (impostors and spoofing attacks). Note that methods called
    within this function will override this parameter and not considered if the
191
    selected criteria is 'min-hter'.
192
    - criteria - the decision threshold criteria ('eer' for EER, 'wer' for
193
    Minimum WER or 'min-hter' for Minimum HTER criteria).
194 195 196 197 198 199 200
  """
    span_min = min(
        numpy.append(licit_neg, spoof_neg)
    )  # the min of the span where we will search for the threshold
    span_max = max(
        numpy.append(licit_pos, spoof_pos)
    )  # the max of the span where we will search for the threshold
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
201 202 203 204 205 206 207
    data = (
        licit_neg,
        licit_pos,
        spoof_neg,
        spoof_pos,
    )  # pack the data into a single list
    return recursive_thr_search(data, span_min, span_max, weight, beta, criteria)
208 209 210 211 212 213 214 215 216 217 218 219 220 221


def epsc_weights(licit_neg, licit_pos, spoof_neg, spoof_pos, points=100):
    """Returns the weights for EPSC

  Keyword arguments:

    - points - number of points to calculate EPSC
  """
    step_size = 1 / float(points)
    weights = numpy.array([(i * step_size) for i in range(points + 1)])
    return weights


Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
222 223 224 225 226 227 228 229 230 231
def epsc_thresholds(
    licit_neg,
    licit_pos,
    spoof_neg,
    spoof_pos,
    points=100,
    criteria="eer",
    omega=None,
    beta=None,
):
232 233 234 235
    """Calculates the optimal thresholds for EPSC, for a range of the weight
    parameter balancing between impostors and spoofing attacks, and for a range
    of the beta parameter balancing between real accesses and all the negatives
    (impostors and spoofing attacks)
236 237 238 239 240 241 242 243

  Keyword arguments:

    - licit_neg - numpy.array of scores for the negatives (licit scenario)
    - licit_pos - numpy.array of scores for the positives (licit scenario)
    - spoof_neg - numpy.array of scores for the negatives (spoof scenario)
    - spoof_pos - numpy.array of scores for the positives (spoof scenario)
    - points - number of points to calculate EPSC
244
    - criteria - the decision threshold criteria ('eer', 'wer' or 'min-hter')
245 246 247 248
    - omega - the value of the parameter omega, balancing between impostors and
    spoofing attacks. If None, it is going to span the full range [0,1].
    Otherwise, can be set to a fixed value or a list of values.
    - beta - the value of the parameter beta, balancing between real accesses
Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
249
    and all the negatives (zero-effort impostors and spoofing attacks). If
250 251
    None, it is going to span the full range [0,1]. Otherwise, can be set to a
    fixed value or a list of values.
252 253 254 255

  """
    step_size = 1 / float(points)

Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
256
    if omega is None:
257
        omega = numpy.array([(i * step_size) for i in range(points + 1)])
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
258 259 260 261 262
    elif (
        not isinstance(omega, list)
        and not isinstance(omega, tuple)
        and not isinstance(omega, numpy.ndarray)
    ):
263 264 265 266
        omega = numpy.array([omega])
    else:
        omega = numpy.array(omega)

Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
267
    if beta is None:
268
        beta = numpy.array([(i * step_size) for i in range(points + 1)])
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
269 270 271 272 273
    elif (
        not isinstance(beta, list)
        and not isinstance(beta, tuple)
        and not isinstance(beta, numpy.ndarray)
    ):
274 275 276 277
        beta = numpy.array([beta])
    else:
        beta = numpy.array(beta)

Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
278
    thresholds = numpy.ndarray([beta.size, omega.size], "float64")
279
    for bindex, b in enumerate(beta):
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
280 281 282 283 284 285 286 287 288
        thresholds[bindex, :] = numpy.array(
            [
                weighted_negatives_threshold(
                    licit_neg, licit_pos, spoof_neg, spoof_pos, w, b, criteria=criteria
                )
                for w in omega
            ],
            "float64",
        )
289 290 291 292 293 294 295 296

    return omega, beta, thresholds


def weighted_err(error_1, error_2, weight):
    """Calculates the weighted error rate between the two input parameters

  Keyword arguments:
297 298
    - error_1 - the first input error rate (FAR for zero effort impostors
    usually)
299 300 301 302 303 304
    - error_2 - the second input error rate (SFAR)
    - weight - the given weight
  """
    return (1 - weight) * error_1 + weight * error_2


Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
305 306 307
def error_rates_at_weight(
    licit_neg, licit_pos, spoof_neg, spoof_pos, omega, threshold, beta=0.5
):
308 309 310
    """Calculates several error rates: FRR, FAR (zero-effort impostors), SFAR,
    FAR_w, HTER_w for a given value of w. It returns the calculated threshold
    as a last argument
311 312 313 314 315 316 317 318

  Keyword arguments:

    - licit_neg - numpy.array of scores for the negatives (licit scenario)
    - licit_pos - numpy.array of scores for the positives (licit scenario)
    - spoof_neg - numpy.array of scores for the negatives (spoof scenario)
    - spoof_pos - numpy.array of scores for the positives (spoof scenario)
    - threshold - the given threshold
319 320 321 322 323
    - omega - the omega parameter balancing between impostors and spoofing
    attacks
    - beta - the weight parameter balancing between real accesses and all the

negative samples (impostors and spoofing attacks).
324 325 326
  """

    farfrr_licit = bob.measure.farfrr(
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
327 328
        licit_neg, licit_pos, threshold
    )  # calculate test frr @ threshold (licit scenario)
329
    farfrr_spoof = bob.measure.farfrr(
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
330 331
        spoof_neg, spoof_pos, threshold
    )  # calculate test frr @ threshold (spoof scenario)
332

Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
333 334
    # we can take this value from farfrr_spoof as well, it doesn't matter
    frr = farfrr_licit[1]
335 336 337 338 339 340 341 342 343 344
    far = farfrr_licit[0]
    sfar = farfrr_spoof[0]

    far_w = weighted_err(far, sfar, omega)
    hter_w = (far_w + frr) / 2
    wer_wb = weighted_err(frr, far_w, beta)

    return (frr, far, sfar, far_w, wer_wb, hter_w, threshold)


Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
345 346 347
def epsc_error_rates(
    licit_neg, licit_pos, spoof_neg, spoof_pos, thresholds, omega, beta
):
Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
348 349
    """Calculates several error rates: FAR_w and WER_wb for the given weights
    (omega and beta) and thresholds (the thresholds need to be computed first
350
    using the method: epsc_thresholds() before passing to this method)
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
    Parameters
    ----------
    licit_neg : array_like
        array of scores for the negatives (licit scenario)
    licit_pos : array_like
        array of scores for the positives (licit scenario)
    spoof_neg : array_like
        array of scores for the negatives (spoof scenario)
    spoof_pos : array_like
        array of scores for the positives (spoof scenario)
    thresholds : array_like
        ndarray with threshold values
    omega : array_like
        array of the omega parameter balancing between impostors
        and spoofing attacks
    beta : array_like
        array of the beta parameter balancing between real accesses
        and all negatives (impostors and spoofing attacks)

    Returns
    -------
    far_w_errors: array_like
        FAR_w
    wer_wb_errors: array_like
        WER_wb
    """
378

Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
379 380
    far_w_errors = numpy.ndarray((beta.size, omega.size), "float64")
    wer_wb_errors = numpy.ndarray((beta.size, omega.size), "float64")
381 382 383

    for bindex, b in enumerate(beta):
        errors = [
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
384 385 386 387 388 389 390 391 392
            error_rates_at_weight(
                licit_neg,
                licit_pos,
                spoof_neg,
                spoof_pos,
                w,
                thresholds[bindex, windex],
                b,
            )
393 394 395 396 397 398 399 400
            for windex, w in enumerate(omega)
        ]
        far_w_errors[bindex, :] = [errors[i][3] for i in range(len(errors))]
        wer_wb_errors[bindex, :] = [errors[i][4] for i in range(len(errors))]

    return far_w_errors, wer_wb_errors


Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
401 402 403
def all_error_rates(
    licit_neg, licit_pos, spoof_neg, spoof_pos, thresholds, omega, beta
):
Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
404 405
    """Calculates several error rates: FAR_w and WER_wb for the given weights
    (omega and beta) and thresholds (the thresholds need to be computed first
406
    using the method: epsc_thresholds() before passing to this method)
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
    Parameters
    ----------
    licit_neg : array_like
        array of scores for the negatives (licit scenario)
    licit_pos : array_like
        array of scores for the positives (licit scenario)
    spoof_neg : array_like
        array of scores for the negatives (spoof scenario)
    spoof_pos : array_like
        array of scores for the positives (spoof scenario)
    thresholds : array_like
        ndarray with threshold values
    omega : array_like
        array of the omega parameter balancing between impostors
        and spoofing attacks
    beta : array_like
        array of the beta parameter balancing between real accesses
        and all negatives (impostors and spoofing attacks)

    Returns
    -------
    far_w_errors: array_like
        FAR_w
    wer_wb_errors: array_like
        WER_wb
    """
434

Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
435 436 437 438 439 440
    frr_errors = numpy.ndarray((beta.size, omega.size), "float64")
    far_errors = numpy.ndarray((beta.size, omega.size), "float64")
    sfar_errors = numpy.ndarray((beta.size, omega.size), "float64")
    far_w_errors = numpy.ndarray((beta.size, omega.size), "float64")
    wer_wb_errors = numpy.ndarray((beta.size, omega.size), "float64")
    hter_wb_errors = numpy.ndarray((beta.size, omega.size), "float64")
441 442 443

    for bindex, b in enumerate(beta):
        errors = [
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
444 445 446 447 448 449 450 451 452
            error_rates_at_weight(
                licit_neg,
                licit_pos,
                spoof_neg,
                spoof_pos,
                w,
                thresholds[bindex, windex],
                b,
            )
453 454 455 456 457 458 459 460 461
            for windex, w in enumerate(omega)
        ]
        frr_errors[bindex, :] = [errors[i][0] for i in range(len(errors))]
        far_errors[bindex, :] = [errors[i][1] for i in range(len(errors))]
        sfar_errors[bindex, :] = [errors[i][2] for i in range(len(errors))]
        far_w_errors[bindex, :] = [errors[i][3] for i in range(len(errors))]
        wer_wb_errors[bindex, :] = [errors[i][4] for i in range(len(errors))]
        hter_wb_errors[bindex, :] = [errors[i][5] for i in range(len(errors))]

Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483
    return (
        frr_errors,
        far_errors,
        sfar_errors,
        far_w_errors,
        wer_wb_errors,
        hter_wb_errors,
    )


def calc_aue(
    licit_neg,
    licit_pos,
    spoof_neg,
    spoof_pos,
    thresholds,
    omega,
    beta,
    l_bound=0,
    h_bound=1,
    var_param="omega",
):
484 485
    """Calculates AUE of EPSC for the given thresholds and weights

486
    Keyword arguments:
487 488 489 490 491 492 493 494

    - licit_neg - numpy.array of scores for the negatives (licit scenario)
    - licit_pos - numpy.array of scores for the positives (licit scenario)
    - spoof_neg - numpy.array of scores for the negatives (spoof scenario)
    - spoof_pos - numpy.array of scores for the positives (spoof scenario)
    - l_bound - lower bound of integration
    - h_bound - higher bound of integration
    - points - number of points to calculate EPSC
495
    - criteria - the decision threshold criteria ('eer', 'wer' or 'min-hter')
Amir MOHAMMADI's avatar
nit  
Amir MOHAMMADI committed
496
    - var_param - name of the parameter which is varied on the abscissa
497
    ('omega' or 'beta')
498 499 500 501
  """

    from scipy import integrate

Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
502 503 504 505
    if var_param == "omega":
        errors = all_error_rates(
            licit_neg, licit_pos, spoof_neg, spoof_pos, thresholds, omega, beta
        )
506 507
        weights = omega  # setting the weights to the varying parameter
    else:
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
508 509 510
        errors = all_error_rates(
            licit_neg, licit_pos, spoof_neg, spoof_pos, thresholds, omega, beta
        )
511 512 513 514 515 516 517
        weights = beta  # setting the weights to the varying parameter

    wer_errors = errors[4].reshape(1, errors[4].size)

    l_ind = numpy.where(weights >= l_bound)[0][0]
    h_ind = numpy.where(weights <= h_bound)[0][-1]
    aue = integrate.cumtrapz(wer_errors, weights)
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
518
    aue = numpy.append([0], aue)  # for indexing purposes, aue is cumulative integration
519 520 521
    aue = aue[h_ind] - aue[l_ind]

    return aue
Amir MOHAMMADI's avatar
Amir MOHAMMADI committed
522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661


def apcer_threshold(desired_apcer, pos, *negs, is_sorted=False):
    """Computes the threshold given the desired APCER as the criteria.

    APCER is computed as max of all APCER_PAI values.
    The threshold will be computed such that the real APCER is **at most** the desired
    value.

    Parameters
    ----------
    desired_apcer : float
        The desired APCER value.
    pos : list
        An array or list of positive scores in float.
    *negs
        A list of negative scores. Each item corresponds to the negative scores of one
        PAI.
    is_sorted : bool, optional
        Set to ``True`` if ALL arrays (pos and negs) are sorted.

    Returns
    -------
    float
        The computed threshold that satisfies the desired APCER.
    """
    threshold = max(
        far_threshold(neg, pos, desired_apcer, is_sorted=is_sorted) for neg in negs
    )
    return threshold


def apcer_bpcer(threshold, pos, *negs):
    """Computes APCER_PAI, APCER, and BPCER given the positive scores and a list of
    negative scores and a threshold.

    Parameters
    ----------
    threshold : float
        The threshold to be used to compute the error rates.
    pos : list
        An array or list of positive scores in float.
    *negs
        A list of negative scores. Each item corresponds to the negative scores of one
        PAI.

    Returns
    -------
    tuple
        A tuple such as (list of APCER_PAI, APCER, BPCER)
    """
    apcers = []
    assert len(negs) > 0, negs
    for neg in negs:
        far, frr = farfrr(neg, pos, threshold)
        apcers.append(far)
    bpcer = frr  # bpcer will be the same in all cases
    return apcers, max(apcers), bpcer


def negatives_per_pai_and_positives(filename, regexps=None, regexp_column="real_id"):
    """Returns scores for Bona-Fide samples and scores for each PAI.
    By default, the real_id column (second column) is used as indication for each
    Presentation Attack Instrument (PAI).

    For example, if you have scores like:
        001 001    bona_fide_sample_1_path 0.9
        001 print  print_sample_1_path     0.6
        001 print  print_sample_2_path     0.6
        001 replay replay_sample_1_path    0.2
        001 replay replay_sample_2_path    0.2
        001 mask   mask_sample_1_path      0.5
        001 mask   mask_sample_2_path      0.5
    this function will return 3 sets of negative scores (for each print, replay, and
    mask PAIs).

    Otherwise, you can provide a list regular expressions that match each PAI.
    For example, if you have scores like:
        001 001      bona_fide_sample_1_path 0.9
        001 print/1  print_sample_1_path     0.6
        001 print/2  print_sample_2_path     0.6
        001 replay/1 replay_sample_1_path    0.2
        001 replay/2 replay_sample_2_path    0.2
        001 mask/1   mask_sample_1_path      0.5
        001 mask/2   mask_sample_2_path      0.5
    and give a list of regexps as ('print', 'replay', 'mask') the function will return 3
    sets of negative scores (for each print, replay, and mask PAIs).


    Parameters
    ----------
    filename : str
        Path to the score file.
    regexps : None, optional
        A list of regular expressions that match each PAI. If not given, the values in
        the real_id column are used to find scores for different PAIs.
    regexp_column : str, optional
        If a list of regular expressions are given, those patterns will be matched
        against the values in this column.

    Returns
    -------
    tuple
        A tuple containing pos scores and a dict of negative scores mapping PAIs to
        their scores.

    Raises
    ------
    ValueError
        If none of the given regular expressions match the values in regexp_column.
    """
    pos = []
    negs = defaultdict(list)
    if regexps:
        regexps = [re.compile(pattern) for pattern in regexps]
        assert regexp_column in ("claimed_id", "real_id", "test_label"), regexp_column

    for claimed_id, real_id, test_label, score in four_column(filename):
        # if it is a Bona-Fide score
        if claimed_id == real_id:
            pos.append(score)
            continue
        if not regexps:
            negs[real_id].append(score)
            continue
        # if regexps is not None or empty and is not a Bona-Fide score
        string = {
            "claimed_id": claimed_id,
            "real_id": real_id,
            "test_label": test_label,
        }[regexp_column]
        for pattern in regexps:
            if pattern.match(string):
                negs[pattern.pattern].append(score)
                break
        else:  # this else is for the for loop: ``for pattern in regexps:``
            raise ValueError(
                f"No regexps: {regexps} match `{string}' from `{regexp_column}' column"
            )
    return pos, negs