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Calculate ranking stats AUROC and AUPRC.
tfp.experimental.substrates.numpy.stats.quantile_auc(
q0,
n0,
q1,
n1,
curve='ROC',
name=None
)
Computes AUROC and AUPRC from quantiles (one for positive trials and one for negative trials).
We use pi(x)
to denote a score. We assume that if pi(x) > k
for some
threshold k
then the event is predicted to be "1", and otherwise it is
predicted to be a "0". Its actual label is y
, which may or may not be the
same.
Area Under Curve: Receiver Operator Characteristic (AUROC) is defined as:
/ 1  TruePositiveRate(k) d FalsePositiveRate(k) / 0
where,
TruePositiveRate(k) = P(pi(x) > k  y = 1)
FalsePositiveRate(k) = P(pi(x) > k  y = 0)
Area Under Curve: PrecisionRecall (AUPRC) is defined as:
/ 1  Precision(k) d Recall(k) / 0
where,
Precision(k) = P(y = 1  pi(x) > k)
Recall(k) = TruePositiveRate(k) = P(pi(x) > k  y = 1)
Notice that AUROC and AUPRC exchange the role of Recall in the integration, i.e.,
Integrand Measure
+++
AUROC  Recall  FPR  +++ AUPRC  Precision  Recall  +++
To learn more about the relationship between AUROC and AUPRC see [1].
Args:
q0
:ND
Tensor
offloat
, Quantiles of predicted probabilities given a negative trial. The firstN1
dimensions are batch dimensions, and the AUC is calculated over the final dimension.n0
:float
or(N1)D Tensor
, Number of negative trials. IfTensor
, dimensions must match the firstN1
dimensions ofq0
.q1
:ND
Tensor
offloat
, Quantiles of predicted probabilities given a positive trial. The firstN1
dimensions are batch dimensions, which must match those ofq0
.n1
:float
or(N1)D Tensor
, Number of positive trials. IfTensor
, dimensions must match the firstN1
dimensions ofq0
.curve
:str
, Specifies the name of the curve to be computed. Must be 'ROC' [default] or 'PR' for the PrecisionRecallcurve.name
:str
, An optional name_scope name.
Returns:
auc
:Tensor
offloat
, area under the ROC or PR curve.
Examples
n = 1000
m = 500
predictions_given_positive = np.random.rand(n)
predictions_given_negative = np.random.rand(m)
q1 = tfp.stats.quantiles(predictions_given_positive, num_quantiles=50)
q0 = tfp.stats.quantiles(predictions_given_negative, num_quantiles=50)
auroc = tfp.stats.quantile_auc(q0, m, q1, n, curve='ROC')
Mathematical Details
The algorithm proceeds by partitioning the combined quantile data into a
series of intervals [a, b)
, approximating the probability mass of
predictions conditioned on a positive trial (d1
) and probability mass of
predictions conditioned on a negative trial (d0
) in each interval [a, b)
,
and accumulating the incremental AUROC/AUPRC as functions of d0
and d1
.
We assume that pi(x) is uniform within a given bucket of each quantile. Thus it will also be uniform within an interval [a, b) as long as the interval does not cross the quantile's bucket boundaries.
A consequence of this assumption is that the cdf is piecewise linear. That is,
P( pi(x) > k  y = 0 ), and, P( pi(x) > k  y = 1 ),
are linear in k
.
Standard AUROC is fairly easier to calculate. Under the conditional uniformity assumptions we have a piece's contribution, [a, b), as:
/ b   P(y = 1  pi > k) d P(pi > k  y = 0)  / a
/ b

=   P(pi > k  y = 1) P(pi = k  y = 0) d k  / a
/ b
1 / (len(q0)  1)  =   P(pi > k  y = 1) d k q0[j + 1]  q0[j]  / a
/ 1
1 / (len(q0)  1)  =  (b  a)  (p1 + u d1) d u q0[j + 1]  q0[j]  / 0
1 / (len(q0)  1) =  (b  a) (p1 + d1 / 2) q0[j + 1]  q0[j]
AUPRC is a bit harder to calculate since the integrand,
P(y > 0  pi(x) > k)
, is conditional on k
rather than a probability over
k
.
We proceed by formulating Precision in terms of the quantiles we have available to us.
Precision(k) = P(y = 1  pi(x) > k)
P(pi(x) > delta  y = 1 ) P(y = 1)
=  P(pi(x) > delta  y = 1 ) P(y = 1) + P(pi(x) > delta  y = 0 ) P(y = 0)
Since the cdf's are piecewise linear, we calculate this piece's contribution to AUPRC by the integral:
/ b   P(y = 1  pi(x) > delta) d P(pi > delta  y = 1)  / a
1 / (len(q1)  1) =  (b  a) * q1[i + 1]  q1[i]
/ 1
 n1 * (u d1 + p1)
*   du
 n1 * (u d1 + p1) + n0 * (u d0 + p0)
/ 0
/ 1
1 / (len(q1)  1)  n1 * (u d1 + p1) =  (b  a)   du q1[i + 1]  q1[i]  n1 * (u d1 + p1) + n0 * (u d0 + p0) / 0
where the equality is a consequence of the piecewise uniformity assumption.
The solution to the integral is given by Mathematica:
Integrate[n1 (u d1 + p1) / (n1 (u d1 + p1) + n0 (u d0 + p0)), {u, 0, 1},
Assumptions > {p1 > 0, d1 > 0, p0 > 0, d0 > 0, n1 > 0, n0 > 0}]
This integral can be solved by hand by noticing that:
f(x) / (f(x) + g(x)) = 1 / (1 + g(x)/f(x))
Thus define: u = 1 + g(x)/f(x) for which: du = [g'(x)h(x)  g(x)f'(x)] / h(x)^2 dx and solving integral 1/u du.
References
[1]: Jesse Davis and Mark Goadrich. The relationship between PrecisionRecall and ROC curves. In International Conference on Machine Learning, 2006. http://dl.acm.org/citation.cfm?id=1143874