scikit-learn - ROC curve with confidence intervals

Question

I am able to get a ROC curve using scikit-learn with fpr, tpr, thresholds = metrics.roc_curve(y_true,y_pred, pos_label=1), where y_true is a list of values based on my gold standard (i.e., 0 for negative and 1 for positive cases) and y_pred is a corresponding list of scores (e.g., 0.053497243, 0.008521122, 0.022781548, 0.101885263, 0.012913795, 0.0, 0.042881547 [...])

I am trying to figure out how to add confidence intervals to that curve, but didn't find any easy way to do that with sklearn.

Solution 1

You can bootstrap the ROC computations (sample with replacement new versions of y_true / y_pred out of the original y_true / y_pred and recompute a new value for roc_curve each time) and the estimate a confidence interval this way.

To take the variability induced by the train test split into account, you can also use the ShuffleSplit CV iterator many times, fit a model on the train split, generate y_pred for each model and thus gather an empirical distribution of roc_curves as well and finally compute confidence intervals for those.

Edit: bootstrapping in python

Here is an example for bootstrapping the ROC AUC score out of the predictions of a single model. I chose to bootstrap the ROC AUC to make it easier to follow as a Stack Overflow answer, but it can be adapted to bootstrap the whole curve instead:

import numpy as np
from scipy.stats import sem
from sklearn.metrics import roc_auc_score

y_pred = np.array([0.21, 0.32, 0.63, 0.35, 0.92, 0.79, 0.82, 0.99, 0.04])
y_true = np.array([0,    1,    0,    0,    1,    1,    0,    1,    0   ])

print("Original ROC area: {:0.3f}".format(roc_auc_score(y_true, y_pred)))

n_bootstraps = 1000
rng_seed = 42  # control reproducibility
bootstrapped_scores = []

rng = np.random.RandomState(rng_seed)
for i in range(n_bootstraps):
    # bootstrap by sampling with replacement on the prediction indices
    indices = rng.randint(0, len(y_pred), len(y_pred))
    if len(np.unique(y_true[indices])) < 2:
        # We need at least one positive and one negative sample for ROC AUC
        # to be defined: reject the sample
        continue

    score = roc_auc_score(y_true[indices], y_pred[indices])
    bootstrapped_scores.append(score)
    print("Bootstrap #{} ROC area: {:0.3f}".format(i + 1, score))

You can see that we need to reject some invalid resamples. However on real data with many predictions this is a very rare event and should not impact the confidence interval significantly (you can try to vary the rng_seed to check).

Here is the histogram:

import matplotlib.pyplot as plt
plt.hist(bootstrapped_scores, bins=50)
plt.title('Histogram of the bootstrapped ROC AUC scores')
plt.show()

Note that the resampled scores are censored in the [0 - 1] range causing a high number of scores in the last bin.

To get a confidence interval one can sort the samples:

sorted_scores = np.array(bootstrapped_scores)
sorted_scores.sort()

# Computing the lower and upper bound of the 90% confidence interval
# You can change the bounds percentiles to 0.025 and 0.975 to get
# a 95% confidence interval instead.
confidence_lower = sorted_scores[int(0.05 * len(sorted_scores))]
confidence_upper = sorted_scores[int(0.95 * len(sorted_scores))]
print("Confidence interval for the score: [{:0.3f} - {:0.3}]".format(
    confidence_lower, confidence_upper))

which gives:

Confidence interval for the score: [0.444 - 1.0]

The confidence interval is very wide but this is probably a consequence of my choice of predictions (3 mistakes out of 9 predictions) and the total number of predictions is quite small.

Another remark on the plot: the scores are quantized (many empty histogram bins). This is a consequence of the small number of predictions. One could introduce a bit of Gaussian noise on the scores (or the y_pred values) to smooth the distribution and make the histogram look better. But then the choice of the smoothing bandwidth is tricky.

Finally as stated earlier this confidence interval is specific to you training set. To get a better estimate of the variability of the ROC induced by your model class and parameters, you should do iterated cross-validation instead. However this is often much more costly as you need to train a new model for each random train / test split.

EDIT: since I first wrote this reply, there is a bootstrap implementation in scipy directly:

https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.bootstrap.html

Solution 2

DeLong Solution [NO bootstrapping]

As some of here suggested, the pROC package in R comes very handy for ROC AUC confidence intervals out-of-the-box, but that packages is not found in python. According to pROC documentation, confidence intervals are calculated via DeLong:

DeLong is an asymptotically exact method to evaluate the uncertainty of an AUC (DeLong et al. (1988)). Since version 1.9, pROC uses the algorithm proposed by Sun and Xu (2014) which has an O(N log N) complexity and is always faster than bootstrapping. By default, pROC will choose the DeLong method whenever possible.

And luckily for us, Yandex Data School has a Fast DeLong implementation on their public repo:

https://github.com/yandexdataschool/roc_comparison

So all credits to them for the DeLong implementation used in this example. So here is how you get a CI via DeLong:

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Created on Tue Nov  6 10:06:52 2018

@author: yandexdataschool

Original Code found in:
https://github.com/yandexdataschool/roc_comparison

updated: Raul Sanchez-Vazquez
"""

import numpy as np
import scipy.stats
from scipy import stats

# AUC comparison adapted from
# https://github.com/Netflix/vmaf/
def compute_midrank(x):
    """Computes midranks.
    Args:
       x - a 1D numpy array
    Returns:
       array of midranks
    """
    J = np.argsort(x)
    Z = x[J]
    N = len(x)
    T = np.zeros(N, dtype=np.float)
    i = 0
    while i < N:
        j = i
        while j < N and Z[j] == Z[i]:
            j += 1
        T[i:j] = 0.5*(i + j - 1)
        i = j
    T2 = np.empty(N, dtype=np.float)
    # Note(kazeevn) +1 is due to Python using 0-based indexing
    # instead of 1-based in the AUC formula in the paper
    T2[J] = T + 1
    return T2


def compute_midrank_weight(x, sample_weight):
    """Computes midranks.
    Args:
       x - a 1D numpy array
    Returns:
       array of midranks
    """
    J = np.argsort(x)
    Z = x[J]
    cumulative_weight = np.cumsum(sample_weight[J])
    N = len(x)
    T = np.zeros(N, dtype=np.float)
    i = 0
    while i < N:
        j = i
        while j < N and Z[j] == Z[i]:
            j += 1
        T[i:j] = cumulative_weight[i:j].mean()
        i = j
    T2 = np.empty(N, dtype=np.float)
    T2[J] = T
    return T2


def fastDeLong(predictions_sorted_transposed, label_1_count, sample_weight):
    if sample_weight is None:
        return fastDeLong_no_weights(predictions_sorted_transposed, label_1_count)
    else:
        return fastDeLong_weights(predictions_sorted_transposed, label_1_count, sample_weight)


def fastDeLong_weights(predictions_sorted_transposed, label_1_count, sample_weight):
    """
    The fast version of DeLong's method for computing the covariance of
    unadjusted AUC.
    Args:
       predictions_sorted_transposed: a 2D numpy.array[n_classifiers, n_examples]
          sorted such as the examples with label "1" are first
    Returns:
       (AUC value, DeLong covariance)
    Reference:
     @article{sun2014fast,
       title={Fast Implementation of DeLong's Algorithm for
              Comparing the Areas Under Correlated Receiver Oerating Characteristic Curves},
       author={Xu Sun and Weichao Xu},
       journal={IEEE Signal Processing Letters},
       volume={21},
       number={11},
       pages={1389--1393},
       year={2014},
       publisher={IEEE}
     }
    """
    # Short variables are named as they are in the paper
    m = label_1_count
    n = predictions_sorted_transposed.shape[1] - m
    positive_examples = predictions_sorted_transposed[:, :m]
    negative_examples = predictions_sorted_transposed[:, m:]
    k = predictions_sorted_transposed.shape[0]

    tx = np.empty([k, m], dtype=np.float)
    ty = np.empty([k, n], dtype=np.float)
    tz = np.empty([k, m + n], dtype=np.float)
    for r in range(k):
        tx[r, :] = compute_midrank_weight(positive_examples[r, :], sample_weight[:m])
        ty[r, :] = compute_midrank_weight(negative_examples[r, :], sample_weight[m:])
        tz[r, :] = compute_midrank_weight(predictions_sorted_transposed[r, :], sample_weight)
    total_positive_weights = sample_weight[:m].sum()
    total_negative_weights = sample_weight[m:].sum()
    pair_weights = np.dot(sample_weight[:m, np.newaxis], sample_weight[np.newaxis, m:])
    total_pair_weights = pair_weights.sum()
    aucs = (sample_weight[:m]*(tz[:, :m] - tx)).sum(axis=1) / total_pair_weights
    v01 = (tz[:, :m] - tx[:, :]) / total_negative_weights
    v10 = 1. - (tz[:, m:] - ty[:, :]) / total_positive_weights
    sx = np.cov(v01)
    sy = np.cov(v10)
    delongcov = sx / m + sy / n
    return aucs, delongcov


def fastDeLong_no_weights(predictions_sorted_transposed, label_1_count):
    """
    The fast version of DeLong's method for computing the covariance of
    unadjusted AUC.
    Args:
       predictions_sorted_transposed: a 2D numpy.array[n_classifiers, n_examples]
          sorted such as the examples with label "1" are first
    Returns:
       (AUC value, DeLong covariance)
    Reference:
     @article{sun2014fast,
       title={Fast Implementation of DeLong's Algorithm for
              Comparing the Areas Under Correlated Receiver Oerating
              Characteristic Curves},
       author={Xu Sun and Weichao Xu},
       journal={IEEE Signal Processing Letters},
       volume={21},
       number={11},
       pages={1389--1393},
       year={2014},
       publisher={IEEE}
     }
    """
    # Short variables are named as they are in the paper
    m = label_1_count
    n = predictions_sorted_transposed.shape[1] - m
    positive_examples = predictions_sorted_transposed[:, :m]
    negative_examples = predictions_sorted_transposed[:, m:]
    k = predictions_sorted_transposed.shape[0]

    tx = np.empty([k, m], dtype=np.float)
    ty = np.empty([k, n], dtype=np.float)
    tz = np.empty([k, m + n], dtype=np.float)
    for r in range(k):
        tx[r, :] = compute_midrank(positive_examples[r, :])
        ty[r, :] = compute_midrank(negative_examples[r, :])
        tz[r, :] = compute_midrank(predictions_sorted_transposed[r, :])
    aucs = tz[:, :m].sum(axis=1) / m / n - float(m + 1.0) / 2.0 / n
    v01 = (tz[:, :m] - tx[:, :]) / n
    v10 = 1.0 - (tz[:, m:] - ty[:, :]) / m
    sx = np.cov(v01)
    sy = np.cov(v10)
    delongcov = sx / m + sy / n
    return aucs, delongcov


def calc_pvalue(aucs, sigma):
    """Computes log(10) of p-values.
    Args:
       aucs: 1D array of AUCs
       sigma: AUC DeLong covariances
    Returns:
       log10(pvalue)
    """
    l = np.array([[1, -1]])
    z = np.abs(np.diff(aucs)) / np.sqrt(np.dot(np.dot(l, sigma), l.T))
    return np.log10(2) + scipy.stats.norm.logsf(z, loc=0, scale=1) / np.log(10)


def compute_ground_truth_statistics(ground_truth, sample_weight):
    assert np.array_equal(np.unique(ground_truth), [0, 1])
    order = (-ground_truth).argsort()
    label_1_count = int(ground_truth.sum())
    if sample_weight is None:
        ordered_sample_weight = None
    else:
        ordered_sample_weight = sample_weight[order]

    return order, label_1_count, ordered_sample_weight


def delong_roc_variance(ground_truth, predictions, sample_weight=None):
    """
    Computes ROC AUC variance for a single set of predictions
    Args:
       ground_truth: np.array of 0 and 1
       predictions: np.array of floats of the probability of being class 1
    """
    order, label_1_count, ordered_sample_weight = compute_ground_truth_statistics(
        ground_truth, sample_weight)
    predictions_sorted_transposed = predictions[np.newaxis, order]
    aucs, delongcov = fastDeLong(predictions_sorted_transposed, label_1_count, ordered_sample_weight)
    assert len(aucs) == 1, "There is a bug in the code, please forward this to the developers"
    return aucs[0], delongcov


alpha = .95
y_pred = np.array([0.21, 0.32, 0.63, 0.35, 0.92, 0.79, 0.82, 0.99, 0.04])
y_true = np.array([0,    1,    0,    0,    1,    1,    0,    1,    0   ])

auc, auc_cov = delong_roc_variance(
    y_true,
    y_pred)

auc_std = np.sqrt(auc_cov)
lower_upper_q = np.abs(np.array([0, 1]) - (1 - alpha) / 2)

ci = stats.norm.ppf(
    lower_upper_q,
    loc=auc,
    scale=auc_std)

ci[ci > 1] = 1

print('AUC:', auc)
print('AUC COV:', auc_cov)
print('95% AUC CI:', ci)

output:

AUC: 0.8
AUC COV: 0.028749999999999998
95% AUC CI: [0.46767194, 1.]

I've also checked that this implementation matches the pROC results obtained from R:

library(pROC)

y_true = c(0,    1,    0,    0,    1,    1,    0,    1,    0)
y_pred = c(0.21, 0.32, 0.63, 0.35, 0.92, 0.79, 0.82, 0.99, 0.04)

# Build a ROC object and compute the AUC
roc = roc(y_true, y_pred)
roc

output:

Call:
roc.default(response = y_true, predictor = y_pred)

Data: y_pred in 5 controls (y_true 0) < 4 cases (y_true 1).
Area under the curve: 0.8

Then

# Compute the Confidence Interval
ci(roc)

output

95% CI: 0.4677-1 (DeLong)

Comments

[Admin]

Thanks for the response. I guess I was hoping to find the equivalent of pROC and have something ready to use. While I know of bootstrapping, I just don't know how I'd go around doing it practically in Python (though there are probably tutorials out there), so was hoping there was a built-in somewhere.

[ogrisel]

Bootstrapping is trivial to implement with numpy.random.random_integer to sample with replacement. I edited the answer.

[ogrisel]

I re-edited my answer as the original had a mistake.

[Waylon Flinn]

edited to use 'randint' instead of 'random_integers' as the latter has been deprecated (and prints 1000 deprecation warnings in jupyter)

[Wassermann]

This gave me different results on my data than R's pROC package. Has anybody else validated this?

[Raul]

@Wassermann, would you mind to provide a reproducible example, I'll be more than happy to check if there is any bug.

[Wassermann]

it won't be that simple as it may seem, but I'll try. However, it will take me some time. I'll let you know.

[Raul]

@Wassermann, I've checked the implementation and I've setup a set of jupyter notebooks in order to make more transparent the reproducibility of my results that can be found in my public repositry here: github.com/RaulSanchezVazquez/…. I've used python version 3.7.5 and R version 3.6.1. Let me know if there is any other thing I can help you with to better understand the discrepancies that you are encountering.

[Wassermann]

after your message I did some more detailed tests on 5 different setups with different OSes, R/Python and various version of packages. It seems that one Python setup (#3 in the linked file) where I use Jupyter gives different results than all other. I did not track it further but my first suspect is scipy ver 1.3.0. Here are csv with test data and my test results: www101.zippyshare.com/v/V1VO0z08/file.html www101.zippyshare.com/v/Nh4q08zM/file.html

[Moreno]

Can you share maybe something that supports this method. I am curious since I had never seen this method before

[jtlz2]

@ogrisel What is sem? Doesn't seem to be used..?

[jtlz2]

@ogrisel Any appetite for plotting the corresponding ROC with uncertainties..? Not sure I have the energy right now :\

[ogrisel]

sem is "standard error of the mean". It's the parametric way to quantify an uncertainty on the mean of a random variable from samples assuming Gaussianity.

[ogrisel]

I am also lazy be matplotlib fill_between is nice for plotting an area of uncertainty: matplotlib.org/3.1.1/api/_as_gen/…

[Kevin]

Almost 10 years late to the party... But is this normal to bootstrap the AUC scores from a single model? I have seen several examples that fit the model to the sampled data, producing the predictions for those samples and bootstrapping the AUC score.

[Admin]

Isn't this a problem as there's non-normality?