"""Calculate metrics."""
from __future__ import annotations
from collections.abc import Iterable
import numpy as np
from scipy import interpolate
from scipy.stats import norm
from sklearn.metrics import confusion_matrix, roc_auc_score, roc_curve
VAR_THRESH = 1e-2
def _div(x: float, y: float, default: float) -> float:
"""Solve the problem of division by 0 and round up.
If y is non-zero, perform x/y and round to 8dp. If it is zero, return the
default.
Parameters
----------
x : float
numerator
y : float
denominator
default : float
return value if y == 0
Returns
-------
division : float
x / y, or default if y == 0
"""
return round(float(x / y), 8) if y != 0 else float(default)
def _tpr_at_fpr(
y_true: Iterable[float],
y_score: Iterable[float],
fpr: float = 0.001,
fpr_perc: bool = False,
) -> float:
"""Compute the TPR at a fixed FPR.
In particular, returns the TPR value corresponding to a particular FPR.
Uses interpolation to fill in gaps.
Parameters
----------
y_true : Iterable[float]
actual class labels
y_score : Iterable[float]
predicted score
fpr : float
false positive rate at which to compute true positive rate
fpr_perc : bool
if the fpr is defined as a percentage
Returns
-------
tpr : float
true positive rate at fpr
"""
if fpr_perc:
fpr /= 100.0
fpr_vals, tpr_vals, thresh_vals = roc_curve(y_true, y_score)
thresh_from_fpr = interpolate.interp1d(fpr_vals, thresh_vals)
tpr_from_thresh = interpolate.interp1d(thresh_vals, tpr_vals)
thresh = thresh_from_fpr(fpr)
return tpr_from_thresh(thresh)
def _expected_auc_var(auc: float, num_pos: int, num_neg: int) -> float:
"""Compute variance of AUC under assumption of uniform probabilities.
Uses the expression given as eqn (2) in https://cs.nyu.edu/~mohri/pub/area.pdf.
Parameters
----------
auc : float
auc value at which to compute the variance
num_pos : int
number of positive examples
num_neg : int
number of negative examples
Returns
-------
var : float
null variance of AUC
"""
p_xxy = p_xyy = 1 / 3
return (
auc * (1 - auc)
+ (num_pos - 1) * (p_xxy - auc**2)
+ (num_neg - 1) * (p_xyy - auc**2)
) / (num_pos * num_neg)
[docs]
def min_max_disc(
y_true: np.ndarray, pred_probs: np.ndarray, x_prop: float = 0.1, log_p: bool = True
) -> tuple[float, float, float, float]:
"""Return non-average-case methods for MIA attacks.
Considers actual frequency of membership amongst samples with highest- and
lowest- assessed probability of membership. If an MIA method confidently
asserts that 5% of samples are members and 5% of samples are not, but
cannot tell for the remaining 90% of samples, then these metrics will flag
this behaviour, but AUC/advantage may not. Since the difference may be
noisy, a p-value against a null of independence of true membership and
assessed membership probability (that is, membership probabilities are
essentially random) is also used as a metric (using a usual Gaussian
approximation to binomial). If the p-value is low and the frequency
difference is high (>0.5) then the MIA attack is successful for some
samples.
Parameters
----------
y : np.ndarray
true labels
yp : np.ndarray
probabilities of labels, or monotonic transformation of probabilities
xprop : float
proportion of samples with highest- and lowest- probability of membership to be
considered
logp : bool
convert p-values to log(p).
Returns
-------
maxd : float
frequency of y=1 amongst proportion xprop of individuals with highest assessed
membership probability
mind : float
frequency of y=1 amongst proportion xprop of individuals with lowest assessed
membership probability
mmd : float
difference between maxd and mind
pval : float
p-value or log-p value corresponding to mmd against null hypothesis that random
variables corresponding to y and yp are independent.
Examples
--------
>>> y = np.random.choice(2, 100)
>>> yp = np.random.rand(100)
>>> maxd, mind, mmd, pval = min_max_desc(y, yp, xprop=0.2, logp=True)
"""
n_examples = int(np.ceil(len(y_true) * x_prop))
pos_frequency = np.mean(y_true) # average frequency
y_order = np.argsort(pred_probs) # ordering permutation
# Frequencies
# y values corresponding to lowest k values of yp
y_lowest_n = y_true[y_order[:n_examples]]
# y values corresponding to highest k values of yp
y_highest_n = y_true[y_order[-(n_examples):]]
maxd = np.mean(y_highest_n)
mind = np.mean(y_lowest_n)
mmd = maxd - mind
# P-value
# mmd is asymptotically distributed as N(0,sdm^2) under null.
sdm = np.sqrt(2 * pos_frequency * (1 - pos_frequency) / n_examples)
pval = 1 - norm.cdf(mmd, loc=0, scale=sdm) # normal CDF
if log_p:
pval = -115.13 if pval < 1e-50 else np.log(pval)
# Return
return maxd, mind, mmd, pval
[docs]
def auc_p_val(auc: float, n_pos: int, n_neg: int) -> tuple[float, float]:
"""Compute the p-value for a given AUC.
Parameters
----------
auc : float
Observed AUC value
n_pos : int
Number of positive examples
n_neg : int
Number of negative examples
Returns
-------
auc_p : float
p-value of observing an AUC > auc by chance
auc_std : float
standard deviation of the NULL AUC diustribution (mean = 0.5)
"""
auc_std = np.sqrt(_expected_auc_var(0.5, n_pos, n_neg))
auc_p = 1 - norm.cdf(auc, loc=0.5, scale=auc_std)
return auc_p, auc_std
def _permute_rows(X_test: np.ndarray, y_test: np.ndarray) -> None:
"""Permute rows.
Parameters
----------
X_test : np.ndarray
Array of features to be permuted.
y_test : np.ndarray
Array of labels to be permuted.
"""
N, _ = X_test.shape
order = np.random.RandomState(seed=10).permutation(N)
X_test = X_test[order, :]
y_test = y_test[order]
[docs]
def get_metrics(
y_pred_proba: np.ndarray, y_test: np.ndarray, permute_rows: bool = True
) -> dict:
"""Calculate metrics, including attacker advantage for MIA binary.
Implemented as Definition 4 on https://arxiv.org/pdf/1709.01604.pdf
which is also implemented in tensorFlow-privacy
https://github.com/tensorflow/privacy.
Parameters
----------
y_test : np.ndarray
Test data labels.
y_pred_proba : np.ndarray of shape [x,2] and type float
Predicted probabilities.
permute_rows : bool, default True
Whether to permute arrays, see: https://github.com/AI-SDC/SACRO-ML/issues/106
Returns
-------
metrics : dict
dictionary of metric values
Notes
-----
Includes the following metrics:
* True positive rate or recall (TPR).
* False positive rate (FPR), proportion of negative examples incorrectly \
classified as positives.
* False alarm rate (FAR), proportion of objects classified as positives \
that are incorrect, also known as false discovery rate.
* True neagative rate (TNR).
* Positive predictive value or precision (PPV).
* Negative predictive value (NPV).
* False neagative rate (FNR).
* Accuracy (ACC).
* F1 Score - harmonic mean of precision and recall.
* Advantage.
"""
if len(y_pred_proba.shape) != 2:
raise ValueError("y_pred must be an array of floats of shape [x,2]")
if y_pred_proba.shape[1] != 2:
raise ValueError("Metrics for multiclass classification are unsupported")
if permute_rows:
_permute_rows(y_pred_proba, y_test)
y_pred = np.argmax(y_pred_proba, axis=1)
y_pred_proba = y_pred_proba[:, 1]
tn, fp, fn, tp = confusion_matrix(y_test, y_pred).ravel()
metrics = {}
# true positive rate or recall
metrics["TPR"] = round(float(tp / (tp + fn)), 8)
# false positive rate, proportion of negative examples
# incorrectly classified as positives
metrics["FPR"] = round(float(fp / (fp + tn)), 8)
# False alarm rate, proportion of things classified as positives
# that are incorrect, also known as false discovery rate
metrics["FAR"] = _div(fp, (fp + tp), 0)
# true negative rate or specificity
metrics["TNR"] = round(float(tn / (tn + fp)), 8)
# precision or positive predictive value
metrics["PPV"] = _div(tp, (tp + fp), 0)
# negative predictive value
metrics["NPV"] = _div(tn, (tn + fn), 0)
# false negative rate
metrics["FNR"] = round(float(fn / (tp + fn)), 8)
# overall accuracy
metrics["ACC"] = round(float((tp + tn) / (tp + fp + fn + tn)), 8)
# harmonic mean of precision and sensitivity
metrics["F1score"] = _div(
2 * metrics["PPV"] * metrics["TPR"], metrics["PPV"] + metrics["TPR"], 0
)
# Advantage
metrics["Advantage"] = float(abs(metrics["TPR"] - metrics["FPR"]))
# calculate AUC of model
metrics["AUC"] = round(roc_auc_score(y_test, y_pred_proba), 8)
# Calculate AUC p-val
metrics["P_HIGHER_AUC"], _ = auc_p_val(
metrics["AUC"], y_test.sum(), len(y_test) - y_test.sum()
)
fmax, fmin, fdif, pdif = min_max_disc(y_test, y_pred_proba)
metrics["FMAX01"] = fmax
metrics["FMIN01"] = fmin
metrics["FDIF01"] = fdif
metrics["PDIF01"] = -pdif # use -log(p) so answer is positive
fmax, fmin, fdif, pdif = min_max_disc(y_test, y_pred_proba, x_prop=0.2)
metrics["FMAX02"] = fmax
metrics["FMIN02"] = fmin
metrics["FDIF02"] = fdif
metrics["PDIF02"] = -pdif # use -log(p) so answer is positive
fmax, fmin, fdif, pdif = min_max_disc(y_test, y_pred_proba, x_prop=0.01)
metrics["FMAX001"] = fmax
metrics["FMIN001"] = fmin
metrics["FDIF001"] = fdif
metrics["PDIF001"] = -pdif # use -log(p) so answer is positive
# Add some things useful for debugging / filtering
metrics["pred_prob_var"] = y_pred_proba.var()
# TPR at various FPR
fpr_vals = [0.5, 0.2, 0.1, 0.01, 0.001, 0.00001]
for fpr in fpr_vals:
tpr = _tpr_at_fpr(y_test, y_pred_proba, fpr=fpr)
name = f"TPR@{fpr}"
metrics[name] = tpr
fpr, tpr, roc_thresh = roc_curve(y_test, y_pred_proba)
# fix for np.inf in first position of roc curves
roc_thresh[0] = 1.0
metrics["fpr"] = fpr
metrics["tpr"] = tpr
metrics["roc_thresh"] = roc_thresh
metrics["n_pos_test_examples"] = y_test.sum()
metrics["n_neg_test_examples"] = (1 - y_test).sum()
return metrics