# import Libraries of other lib packages import bob.math import bob.io.base from ._library import * from . import version from .version import module as __version__ from . import plot from . import calibration from . import load import numpy def fprfnr(negatives, positives, threshold): """Alias for :py:func:bob.measure.farfrr""" return farfrr(negatives, positives, threshold) def mse (estimation, target): """Mean square error between a set of outputs and target values Uses the formula: .. math:: MSE(\hat{\Theta}) = E[(\hat{\Theta} - \Theta)^2] Estimation (:math:\hat{\Theta}) and target (:math:\Theta) are supposed to have 2 dimensions. Different examples are organized as rows while different features in the estimated values or targets are organized as different columns. Parameters: estimation (array): an N-dimensional array that corresponds to the value estimated by your procedure target (array): an N-dimensional array that corresponds to the expected value Returns: float: The average of the squared error between the estimated value and the target """ return numpy.mean((estimation - target)**2, 0) def rmse (estimation, target): """Calculates the root mean square error between a set of outputs and target Uses the formula: .. math:: RMSE(\hat{\Theta}) = \sqrt(E[(\hat{\Theta} - \Theta)^2]) Estimation (:math:\hat{\Theta}) and target (:math:\Theta) are supposed to have 2 dimensions. Different examples are organized as rows while different features in the estimated values or targets are organized as different columns. Parameters: estimation (array): an N-dimensional array that corresponds to the value estimated by your procedure target (array): an N-dimensional array that corresponds to the expected value Returns: float: The square-root of the average of the squared error between the estimated value and the target """ return numpy.sqrt(mse(estimation, target)) def relevance (input, machine): """Calculates the relevance of every input feature to the estimation process Uses the formula: Neural Triggering System Operating on High Resolution Calorimetry Information, Anjos et al, April 2006, Nuclear Instruments and Methods in Physics Research, volume 559, pages 134-138 .. math:: R(x_{i}) = |E[(o(x) - o(x|x_{i}=E[x_{i}]))^2]| In other words, the relevance of a certain input feature **i** is the change on the machine output value when such feature is replaced by its mean for all input vectors. For this to work, the input parameter has to be a 2D array with features arranged column-wise while different examples are arranged row-wise. Parameters: input (array): an N-dimensional array that corresponds to the value estimated by your model machine (object): A machine that can be called to "process" your input Returns: array: An 1D float array as large as the number of columns (second dimension) of your input array, estimating the "relevance" of each input column (or feature) to the score provided by the machine. """ o = machine(input) i2 = input.copy() retval = numpy.ndarray((input.shape[1],), 'float64') retval.fill(0) for k in range(input.shape[1]): i2[:,:] = input #reset i2[:,k] = numpy.mean(input[:,k]) retval[k] = (mse(machine(i2), o).sum())**0.5 return retval def recognition_rate(cmc_scores, threshold = None, rank = 1): """Calculates the recognition rate from the given input It is identical to the CMC value for the given rank. The input has a specific format, which is a list of two-element tuples. Each of the tuples contains the negative :math:\\{S_p^-\\} and the positive :math:\\{S_p^+\\} scores for one probe item :math:p, or None in case of open set recognition. If threshold is set to None, the rank 1 recognition rate is defined as the number of test items, for which the highest positive :math:\\max\\{S_p^+\\} score is greater than or equal to all negative scores, divided by the number of all probe items :math:P: .. math:: \\mathrm{RR} = \\frac{1}{P} \\sum_{p=1}^{P} \\begin{cases} 1 & \\mathrm{if } \\max\\{S_p^+\\} >= \\max\\{S_p^-\\}\\\\ 0 & \\mathrm{otherwise} \\end{cases} For a given rank :math:r>1, up to :math:r negative scores that are higher than the highest positive score are allowed to still count as correctly classified in the top :math:r rank. If threshold :math:\\theta is given, **all** scores below threshold will be filtered out. Hence, if all positive scores are below threshold :math:\\max\\{S_p^+\\} < \\theta, the probe will be misclassified **at any rank**. For open set recognition, i.e., when there exist a tuple including negative scores without corresponding positive scores (None), and **all** negative scores are below threshold :math:\\max\\{S_p^+\\} < \\theta, the probe item is correctly rejected, **and it does not count into the denominator** :math:P. When no threshold is provided, the open set probes will **always** count as misclassified, regardless of the rank. .. warn: For open set tests, this rate does not correspond to a standard rate. Please use :py:func:detection_identification_rate and :py:func:false_alarm_rate instead. Parameters: cmc_scores (:py:class:list): A list in the format [(negatives, positives), ...] containing the CMC scores (i.e. :py:class:list: A list of tuples, where each tuple contains the negative and positive scores for one probe of the database). Each pair contains the negative and the positive scores for **one probe item**. Each pair can contain up to one empty array (or None), i.e., in case of open set recognition. threshold (:obj:float, optional): Decision threshold. If not None, **all** scores will be filtered by the threshold. In an open set recognition problem, all open set scores (negatives with no corresponding positive) for which all scores are below threshold, will be counted as correctly rejected and **removed** from the probe list (i.e., the denominator). rank (:obj:int, optional): The rank for which the recognition rate should be computed, 1 by default. Returns: float: The (open set) recognition rate for the given rank, a value between 0 and 1. """ # If no scores are given, the recognition rate is exactly 0. if not cmc_scores: return 0. correct = 0 counter = 0 for neg, pos in cmc_scores: # set all values that are empty before to None if pos is not None and not numpy.array(pos).size: pos = None if neg is not None and not numpy.array(neg).size: neg = None if pos is None and neg is None: raise ValueError("One pair of the CMC scores has neither positive nor negative values") # filter out any negative or positive scores below threshold; scores with exactly the threshold are also filtered out # now, None and an empty array have different meanings. if threshold is not None: if neg is not None: neg = numpy.array(neg)[neg > threshold] if pos is not None: pos = numpy.array(pos)[pos > threshold] if pos is None: # no positives, so we definitely do not have a match; # check if we have negatives above threshold if not neg.size: # we have no negative scores over the threshold, so we have correctly rejected the probe # don't increase any of the two counters... continue # we have negatives over threshold, so we have incorrect classifications; independent on the actual rank counter += 1 else: # we have a positive, so we need to count the probe counter += 1 if not numpy.array(pos).size: # all positive scores have been filtered out by the threshold, we definitely have a mis-match continue # get the maximum positive score for the current probe item # (usually, there is only one positive score, but just in case...) max_pos = numpy.max(pos) if neg is None or not numpy.array(neg).size: # if we had no negatives, or all negatives were below threshold, we have a match at rank 1 correct += 1 else: # count the number of negative scores that are higher than the best positive score index = numpy.sum(neg >= max_pos) if index < rank: correct += 1 return float(correct) / float(counter) def cmc(cmc_scores): """Calculates the cumulative match characteristic (CMC) from the given input. The input has a specific format, which is a list of two-element tuples. Each of the tuples contains the negative and the positive scores for one probe item. For each probe item the probability that the rank :math:r of the positive score is calculated. The rank is computed as the number of negative scores that are higher than the positive score. If several positive scores for one test item exist, the **highest** positive score is taken. The CMC finally computes how many test items have rank r or higher, divided by the total number of test values. .. note:: The CMC is not available for open set classification. Please use the :py:func:detection_identification_rate and :py:func:false_alarm_rate instead. Parameters ---------- cmc_scores : :py:class:list A list in the format [(negatives, positives), ...] containing the CMC scores. Each pair contains the negative and the positive scores for **one probe item**. Each pair can contain up to one empty array (or None), i.e., in case of open set recognition. Returns ------- 1D :py:class:numpy.ndarray of float A 1D float array representing the CMC curve. The rank 1 recognition rate can be found in array[0], rank 2 rate in array[1], and so on. The number of ranks (array.shape[0]) is the number of gallery items. Values are in range [0,1]. """ # If no scores are given, we cannot plot anything probe_count = float(len(cmc_scores)) if not probe_count: raise ValueError("The given set of scores is empty") # compute MC match_characteristic = numpy.zeros((max([len(neg) for neg, _ in cmc_scores if neg is not None])+1,), numpy.int) for neg, pos in cmc_scores: if pos is None or not numpy.array(pos).size: raise ValueError("For the CMC computation at least one positive score per pair is necessary.") if neg is None: neg = [] # get the maximum positive score for the current probe item # (usually, there is only one positive score, but just in case...) max_pos = numpy.max(pos) # count the number of negative scores that are higher than the best positive score index = numpy.sum(neg >= max_pos) match_characteristic[index] += 1 # cumulate cumulative_match_characteristic = numpy.cumsum(match_characteristic, dtype=numpy.float64) return cumulative_match_characteristic / probe_count def detection_identification_rate(cmc_scores, threshold, rank = 1): """Computes the detection and identification rate for the given threshold. This value is designed to be used in an open set identification protocol, and defined in Chapter 14.1 of [LiJain2005]_. Although the detection and identification rate is designed to be computed on an open set protocol, it uses only the probe elements, for which a corresponding gallery element exists. For closed set identification protocols, this function is identical to :py:func:recognition_rate. The only difference is that for this function, a threshold for the scores need to be defined, while for :py:func:recognition_rate it is optional. Parameters: cmc_scores (:py:class:list): A list in the format [(negatives, positives), ...] containing the CMC. Each pair contains the negative and the positive scores for **one probe item**. Each pair can contain up to one empty array (or None), i.e., in case of open set recognition. threshold (float): The decision threshold :math:\\tau. rank (:obj:int, optional): The rank for which the curve should be plotted Returns: float: The detection and identification rate for the given threshold. """ # count the correctly classifier probes correct = 0 counter = 0 for neg, pos in cmc_scores: if pos is None or not numpy.array(pos).size: # we only consider probes with corresponding gallery items continue # we have an in-gallery probe counter += 1 # check, if it is correctly classified if neg is None: neg = [] # get the maximum positive score for the current probe item # (usually, there is only one positive score, but just in case...) max_pos = numpy.max(pos) index = numpy.sum(neg >= max_pos) # compute the rank (in fact, rank - 1) if max_pos >= threshold and index < rank: correct += 1 if not counter: logger.warn("No in-gallery probe was found") return 0. return float(correct) / float(counter) def false_alarm_rate(cmc_scores, threshold): """Computes the false alarm rate for the given threshold,. This value is designed to be used in an open set identification protocol, and defined in Chapter 14.1 of [LiJain2005]_. The false alarm rate is designed to be computed on an open set protocol, it uses only the probe elements, for which **no** corresponding gallery element exists. Parameters: cmc_scores (:py:class:list): A list in the format [(negatives, positives), ...] containing the CMC scores (i.e. :py:class:list: A list of tuples, where each tuple contains the negative and positive scores for one probe of the database). Each pair contains the negative and the positive scores for **one probe item**. Each pair can contain up to one empty array (or None), i.e., in case of open set recognition. threshold (float): The decision threshold :math:\\tau. Returns: float: The false alarm rate. """ incorrect = 0 counter = 0 for neg, pos in cmc_scores: # we only consider the out-of-gallery probes, i.e., with no positive scores if pos is None or not numpy.array(pos).size: counter += 1 # check if the probe is above threshold if neg is None or not numpy.array(neg).size: raise ValueError("One pair of the CMC scores has neither positive nor negative values") if numpy.max(neg) >= threshold: incorrect += 1 if not counter: logger.warn("No out-of-gallery probe was found") return 0. return float(incorrect) / float(counter) def eer(negatives, positives, is_sorted=False, also_farfrr=False): """Calculates the Equal Error Rate (EER). Please note that it is possible that eer != fpr != fnr. This function returns (fpr + fnr) / 2 as eer. If you also need the fpr and fnr values, set also_farfrr to True. Parameters ---------- negatives : array_like (1D, float) The scores for comparisons of objects of different classes. positives : array_like (1D, float) The scores for comparisons of objects of the same class. is_sorted : bool Are both sets of scores already in ascendantly sorted order? also_farfrr : bool If True, it will also return far and frr. Returns ------- eer : float The Equal Error Rate (EER). fpr : float The False Positive Rate (FPR). Returned only when also_farfrr is True. fnr : float The False Negative Rate (FNR). Returned only when also_farfrr is True. """ threshold = eer_threshold(negatives, positives, is_sorted) far, frr = farfrr(negatives, positives, threshold) if also_farfrr: return (far + frr) / 2.0, far, frr return (far + frr) / 2.0 def get_config(): """Returns a string containing the configuration information. """ import bob.extension return bob.extension.get_config(__name__, version.externals) # gets sphinx autodoc done right - don't remove it __all__ = [_ for _ in dir() if not _.startswith('_')]