Area under the receiver operating curve Area under the receiver operating curve Copyright 2019 Allen Downey License: http://creativecommons.org/licenses/by/4.0/ (http://creativecommons.org/licenses/by/4.0/)

Area under ROC Area under ROC As a way of understanding AUC ROC, let's look at the relationship between AUC and Cohen's effect size. Cohen's effect size, d , expresses the difference between two groups as the number of standard deviations between the means. As d increases, we expect it to be easier to distinguish between groups, so we expect AUC to increase. I'll start in one dimension and then generalize to multiple dimensions.a

Here are the means and standard deviations for two hypothetical groups. In [2]: mu1 = 0 sigma = 1 d = 1 mu2 = mu1 + d;

I'll generate two random samples with these parameters. In [3]: n = 1000 sample1 = np.random.normal(mu1, sigma, n) sample2 = np.random.normal(mu2, sigma, n);

If we put a threshold at the midpoint between the means, we can compute the fraction of Group 0 that would be above the threshold. I'll call that the false positive rate. In [4]: thresh = (mu1 + mu2) / 2 np.mean(sample1 > thresh) Out[4]: 0.301

And here's the fraction of Group 1 that would be below the threshold, which I'll call the false negative rate. In [5]: np.mean(sample2 < thresh) Out[5]: 0.325

Plotting misclassi cation Plotting misclassi cation To see what these overlapping distributions look like, I'll plot a kernel density estimate (KDE). In [8]: def plot_misclassification(sample1, sample2, thresh): """Plot KDEs and shade the areas of misclassification. sample1: sequence sample2: sequence thresh: number """ kde1 = make_kde(sample1) clipped = kde1[kde1.index>=thresh] plot_kde(kde1, clipped, 'C0') kde2 = make_kde(sample2) clipped = kde2[kde2.index<=thresh] plot_kde(kde2, clipped, 'C1')

Here's what it looks like with the threshold at 0. There are many false positives, shown in blue, and few false negatives, in orange. In [9]: plot_misclassification(sample1, sample2, 0)

With a higher threshold, we get fewer false positives, at the cost of more false negatives. In [10]: plot_misclassification(sample1, sample2, 1)

The receiver operating curve The receiver operating curve The receiver operating curve (ROC) represents this tradeoff. To plot the ROC, we have to compute the false positive rate (which we saw in the gure above), and the true positive rate (not shown in the gure). The following function computes these metrics. In [11]: def fpr_tpr(sample1, sample2, thresh): """Compute false positive and true positive rates. sample1: sequence sample2: sequence thresh: number returns: tuple of (fpr, tpf) """ fpr = np.mean(sample1>thresh) tpr = np.mean(sample2>thresh) return fpr, tpr

When the threshold is high, the false positive rate is low, but so is the true positive rate. In [12]: fpr_tpr(sample1, sample2, 1) As we decrease the threshold, the true positive rate increases, but so does the false positive rate. In [13]: fpr_tpr(sample1, sample2, 0) Out[12]: (0.15, 0.501) Out[13]: (0.483, 0.846)

The ROC shows this tradeoff over a range of thresholds. I sweep thresholds from high to low so the ROC goes from left to right. In [14]: from scipy.integrate import trapz def plot_roc(sample1, sample2, label): """Plot the ROC curve and return the AUC. sample1: sequence sample2: sequence label: string returns: AUC """ threshes = np.linspace(5, -3) roc = [fpr_tpr(sample1, sample2, thresh) for thresh in threshes] fpr, tpr = np.transpose(roc) plt.plot(fpr, tpr, label=label) plt.xlabel('False positive rate') plt.ylabel('True positive rate') auc = trapz(tpr, fpr) return auc

Here's the ROC for the samples. With d=1 , the area under the curve is about 0.75. That might be a good number to remember. In [15]: auc = plot_roc(sample1, sample2, '') Out[15]: 0.761592

Now let's see what that looks like for a range of d .

In [16]: mu1 = 0 sigma = 1 n = 1000 res = [] for mu2 in [3, 2, 1.5, 0.75, 0.25]: sample1 = np.random.normal(mu1, sigma, n) sample2 = np.random.normal(mu2, sigma, n) d = (mu2-mu1) / sigma label = 'd = %0.2g' % d auc = plot_roc(sample1, sample2, label) res.append((d, auc)) plt.legend();

This function computes AUC as a function of d . In [17]: def plot_auc_vs_d(res, label): d, auc = np.transpose(res) plt.plot(d, auc, label=label, alpha=0.8) plt.xlabel('Cohen effect size') plt.ylabel('Area under ROC')

The following gure shows AUC as a function of d . In [18]: plot_auc_vs_d(res, '') Not suprisingly, AUC increases as d increases.

Multivariate distributions Multivariate distributions Now let's see what happens if we have more than one variable, with a difference in means along more than one dimension. First, I'll generate a 2-D sample with d=1 along both dimensions. In [19]: from scipy.stats import multivariate_normal d = 1 mu1 = [0, 0] mu2 = [d, d] rho = 0 sigma = [[1, rho], [rho, 1]] In [20]: sample1 = multivariate_normal(mu1, sigma).rvs(n) sample2 = multivariate_normal(mu2, sigma).rvs(n); Out[19]: [[1, 0], [0, 1]]

The mean of sample1 should be near 0 for both features. In [21]: np.mean(sample1, axis=0) And the mean of sample2 should be near 1. In [22]: np.mean(sample2, axis=0) Out[21]: array([ 0.01204411, -0.05193738]) Out[22]: array([0.97947675, 1.02358947])

The following scatterplot shows what this looks like in 2-D. In [23]: x, y = sample1.transpose() plt.plot(x, y, '.', alpha=0.3) x, y = sample2.transpose() plt.plot(x, y, '.', alpha=0.3) plt.xlabel('X') plt.ylabel('Y') plt.title('Scatter plot for samples with d=1 in both dimensions');

Some points are clearly classi able, but there is substantial overlap in the distributions.

We can see the same thing if we estimate a 2-D density function and make a contour plot. In [25]: X, Y, Z = kde_scipy(sample1) plt.contour(X, Y, Z, cmap=plt.cm.Blues, alpha=0.7) X, Y, Z = kde_scipy(sample2) plt.contour(X, Y, Z, cmap=plt.cm.Oranges, alpha=0.7) plt.xlabel('X') plt.ylabel('Y') plt.title('KDE for samples with d=1 in both dimensions');

Classi cation with logistic regression Classi cation with logistic regression To see how distinguishable the samples are, I'll use logistic regression. To get the data into the right shape, I'll make two DataFrames, label them, concatenate them, and then extract the labels and the features.

In [26]: df1 = pd.DataFrame(sample1) df1['label'] = 1 df1.describe() In [27]: df1[[0,1]].corr() Out[26]: 0 1 label count 1000.000000 1000.000000 1000.0 mean 0.012044 -0.051937 1.0 std 0.971861 0.976814 0.0 min -3.580857 -3.061129 1.0 25% -0.596927 -0.696824 1.0 50% 0.071937 -0.044057 1.0 75% 0.655457 0.615113 1.0 max 3.053507 3.292066 1.0 Out[27]: 0 1 0 1.000000 0.021376 1 0.021376 1.000000

In [28]: df2 = pd.DataFrame(sample2) df2['label'] = 2 df2.describe() In [29]: df2[[0,1]].corr() Out[28]: 0 1 label count 1000.000000 1000.000000 1000.0 mean 0.979477 1.023589 2.0 std 0.983136 0.967058 0.0 min -2.231272 -2.027548 2.0 25% 0.291482 0.417082 2.0 50% 1.008545 1.008277 2.0 75% 1.670930 1.647037 2.0 max 3.869119 4.138071 2.0 Out[29]: 0 1 0 1.00000 -0.04433 1 -0.04433 1.00000

In [30]: df = pd.concat([df1, df2], ignore_index=True) df.label.value_counts() Out[30]: 2 1000 1 1000 Name: label, dtype: int64

X is the array of features; y is the vector of labels. In [31]: X = df[[0, 1]] y = df.label; Now we can t the model. In [32]: from sklearn.linear_model import LogisticRegression model = LogisticRegression(solver='lbfgs').fit(X, y); And compute the AUC. In [33]: from sklearn.metrics import roc_auc_score y_pred_prob = model.predict_proba(X)[:,1] auc = roc_auc_score(y, y_pred_prob) With two features, we can do better than with just one. Out[33]: 0.853391

AUC as a function of rho AUC as a function of rho The following function contains the code from the previous section, with rho as a parameter.

In [34]: def multivariate_normal_auc(d, rho=0): """Generate multivariate normal samples and classify them. d: Cohen's effect size along each dimension num_dims: number of dimensions returns: AUC """ mu1 = [0, 0] mu2 = [d, d] sigma = [[1, rho], [rho, 1]] # generate the samples sample1 = multivariate_normal(mu1, sigma).rvs(n) sample2 = multivariate_normal(mu2, sigma).rvs(n) # label the samples and extract the features and labels df1 = pd.DataFrame(sample1) df1['label'] = 1 df2 = pd.DataFrame(sample2) df2['label'] = 2 df = pd.concat([df1, df2], ignore_index=True) X = df.drop(columns='label') y = df.label # run the model model = LogisticRegression(solver='lbfgs').fit(X, y) y_pred_prob = model.predict_proba(X)[:,1] # compute AUC auc = roc_auc_score(y, y_pred_prob) return auc

In [35]: res = [(rho, multivariate_normal_auc(d=1, rho=rho)) for rho in np.linspace(-0.9, 0.9)] rhos, aucs = np.transpose(res) plt.plot(rhos, aucs) plt.xlabel('Correlation (rho)') plt.ylabel('Area under ROC') plt.title('AUC as a function of correlation');

AUC as a function of d AUC as a function of d The following function contains the code from the previous section, generalized to handle more than 2 dimensions.

In [36]: def multivariate_normal_auc(d, num_dims=2): """Generate multivariate normal samples and classify them. d: Cohen's effect size along each dimension num_dims: number of dimensions returns: AUC """ # compute the mus mu1 = np.zeros(num_dims) mu2 = np.full(num_dims, d) # and sigma sigma = np.identity(num_dims) # generate the samples sample1 = multivariate_normal(mu1, sigma).rvs(n) sample2 = multivariate_normal(mu2, sigma).rvs(n) # label the samples and extract the features and labels df1 = pd.DataFrame(sample1) df1['label'] = 1 df2 = pd.DataFrame(sample2) df2['label'] = 2 df = pd.concat([df1, df2], ignore_index=True) X = df.drop(columns='label') y = df.label # run the model model = LogisticRegression(solver='lbfgs').fit(X, y) y_pred_prob = model.predict_proba(X)[:,1] # compute AUC

Con rming what we have seen before: In [37]: multivariate_normal_auc(d=1, num_dims=1) In [38]: multivariate_normal_auc(d=1, num_dims=2) Out[37]: 0.7680769999999999 Out[38]: 0.8322470000000001

Now we can sweep a range of effect sizes. In [39]: def compute_auc_vs_d(num_dims): """Sweep a range of effect sizes and compute AUC. num_dims: number of dimensions returns: list of """ effect_sizes = np.linspace(0, 4) return [(d, multivariate_normal_auc(d, num_dims)) for d in effect_sizes] In [40]: res1 = compute_auc_vs_d(1) res2 = compute_auc_vs_d(2) res3 = compute_auc_vs_d(3) res4 = compute_auc_vs_d(4);

And plot the results. In [41]: plot_auc_vs_d(res4, 'num_dim=4') plot_auc_vs_d(res3, 'num_dim=3') plot_auc_vs_d(res2, 'num_dim=2') plot_auc_vs_d(res1, 'num_dim=1') plt.title('AUC vs d for different numbers of features') plt.legend(); With more features, the AUC gets better, assuming the features are independent.

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