# Average Precision#

Average precision (AP) summarizes a precision-recall (PR) curve into a single value representing the average of all precisions. It is generally understood as the approximation of the area under the PR curve. AP ranges between 0 and 1, where a perfect model has precision, recall, and AP scores of 1. The larger the metric, the better a model performs across different thresholds.

Guides: Precision and Recall

Read the precision and the recall guides if you're not familiar with those metrics.

Unlike metrics like precision, recall, and F_{1}-score, which
are threshold-dependent where a confidence threshold value must be defined to compute them, AP is a key performance
**threshold-independent** metric that removes the dependency of selecting one confidence threshold value and measures a
model's performance across all thresholds.

AP is commonly used to evaluate the performance of object detection and information retrieval workflows. This metric (or an aggregated version of it called mean average precision (mAP)) is the primary metric used across popular object detection benchmarks such as PASCAL VOC 2012, COCO, and Open Images V7.

## Implementation Details#

The general definition of AP is finding the approximation of the area under the PR curve. The actual area under the curve, where \(p(r)\) is the precision at recall \(r\), can be defined:

The integral above is in practice replaced with a **finite sum** over every **unique recall** value (or over a set of
evenly spaced recall values) — different interpolation methods are discussed in the section below. The average precision
over a set of recall values or over a range of thresholds at which we are evaluating the model can be defined:

where

- \(p(k)\) is the precision at threshold \(k\)
- \(r(k)\) is the recall at threshold \(k\)
- \(n\) is the number of thresholds

Let’s take a closer look at different implementations of the AP metric. Two primary machine learning workflows that
use AP as a main evaluation metric are **object detection** and **information retrieval**, but AP is implemented
slightly differently for both.

### Object Detection#

Let’s consider the following simple example:

The above three images show a total of four ground truth objects, all of which are matched with an inference bounding box based on the Intersection over Union (IoU) scores. Let’s look at each inference bounding box and sort them by their confidence score in descending order.

Inference | TP/FP | cumsum(TP) | cumsum(FP) | Precision | Recall | |
---|---|---|---|---|---|---|

H | 0.99 | TP | 1 | 0 | 1.0 | 0.2 |

B | 0.88 | TP | 2 | 0 | 1.0 | 0.4 |

E | 0.72 | FP | 2 | 1 | 0.667 | 0.4 |

A | 0.70 | FP | 2 | 2 | 0.5 | 0.4 |

J | 0.54 | FP | 2 | 3 | 0.4 | 0.4 |

D | 0.54 | TP | 3 | 3 | 0.5 | 0.6 |

I | 0.38 | TP | 4 | 3 | 0.571 | 0.8 |

C | 0.2 | FP | 4 | 4 | 0.5 | 0.8 |

F | 0.2 | FP | 4 | 5 | 0.444 | 0.8 |

G | 0.1 | TP | 5 | 5 | 0.5 | 1.0 |

Guides: TP/FP Counts in Object Detection

Read the Intersection over Union (IoU), the Geometry Matching, and the TP / FP / FN / TN guides if you're not familiar with those terminologies.

In order to compute AP, we first need to define precision and recall at each threshold. In this example, we are going to use every unique confidence score as threshold to calculate precision and recall metrics, so we have the complete list of unique recall values. Starting from the top, each inference is assigned to be either a true positive (TP) or false positive (FP) depending on the matching results — if the inference is matched with a ground truth, then it's a TP; otherwise, a FP. Notice that all inferences in this table are considered to be positive (either a TP or FP) because we are evaluating them at the thresholds equal to their confidence scores. Then, the cumulative sum of TP and FP counts respectively from the previous rows are computed at each row. Using these accumulated TP and FP counts, the precision and recall metrics can be defined at each threshold. We are using the cumulative sum because once again each row is evaluated at the threshold equal to its confidence score, so only the upper rows and the current row (i.e., inference with score greater than or equal to the threshold) count as positive inferences.

Now that we have the precision and recall defined at each threshold, let’s plot the PR curve:

Notice the zigzag pattern, often referred to as “**wiggles**” — the precision goes down with FPs and goes up again with TPs as the
recall increases. It is a common practice to first smooth out the wiggles before calculating the AP metric by taking the
maximum precision value to its graphical right side of each recall value. This is why AP is called the **approximation**
of the area under the PR curve. The interpolated precision at each recall is defined:

The PR curve is re-plotted using the interpolated precisions (see orange line in the plot below).

The precisions (y-values) of the smoothed out curve, the orange line on the plot above, are
**monotonically decreasing**. We’re now ready to calculate AP, which is simply the **area under the smoothed out curve**:

The start and the end of PR curve

Notice the above PR curve doesn't start at zero recall. It is because there is no valid threshold that will result in zero recall. In order to ensure that the graph starts on the y-axis, the first point on the curve extends all the way to the y-axis. Similarly, the end of the PR curve doesn't always extend all the way to the recall value of 1. This is because not all the ground truths are matched. Unlike the start of the curve, the tail of the curve doesn't get extended when calculating AP.

Smoothing

Although smoothing is considered as the standard implementation of average precision, scikit-learn's average precision implementation does not smooth out the precisions as mentioned above.

The example above computes AP at **all unique recall** values whenever the maximum precision value drops. This is the
most precise implementation of the metric, used in popular benchmarks like the
PASCAL VOC challenge since 2010.
Prior to 2010, the PASCAL VOC challenge had a different implementation for the AP calculation where the **11 linearly
spaced recall values** from 0.0 to 1.0 were used instead of all unique recall values.

#### 11-point Interpolation#

The 11-point interpolated AP was used in the PASCAL VOC until a new AP calculation, what's considered as the standard now, which was adopted in 2010. This interpolation uses the average of the maximum precision values for 11 linearly spaced recall values from 0.0 to 1.0:

When the precisions at certain recall values become extremely small, they are exempted from the AP calculation. The intention of using this 11-point interpolation, according to the original paper, is as follows:

The intention in interpolating the precision/recall curve in this way is to reduce the impact of the “wiggles” in the precision/recall curve, caused by small variations in the ranking of examples.

However, this linearly interpolated method suffers from being less precise and not being able to measure the difference
with low APs due to the approximation mentioned above. The COCO benchmark
uses a linear interpolation method but with **101** recall values.

### Information Retrieval#

Information retrieval is a machine learning workflow where the user provides a query, and the model returns a score that measures how similar each data is to the query to find the most relevant information from the database.

Average precision is one of the popular metrics used in information retrieval workflow along with object detection workflow. However, the implementation used in information retrieval workflow is slightly different than the one from the section above. The formula of the metric is defined:

where

- \(n\) is the total number of data that you are interested in
- \(\text{GTP}\) is the total number of positive ground truths
- \(p(k)\) is the precision at rank \(k\) data
- \(rel(k)\) is the relevance at rank \(k\) data (1 if the data is relevant, 0 otherwise)

Let’s consider the following example of retrieving similar images to the query from a database of images with different shapes and colors:

The retrieved images are the complete list of images from the database that are ranked by their similarity scores, which are predicted from the model, where the left-most image is the most similar to the query image.

From the retrieved images, the ones with a circle are the TPs, where \(rel(k) = 1\), and any other shapes are labeled as FPs, where \(rel(k) = 0\). Then by simply accumulating all the counts of the TPs in each rank, \(p(k) \times rel(k)\) can be calculated at each rank.

AP is the sum of all the relevant precisions over the total number of positive samples in the database, so in this example, AP becomes:

### Mean Average Precision (mAP)#

The mean average precision (mAP) is simply the macro-average of the AP calculated across different classes for object detection workflow or across different queries for information retrieval workflow. It is important to note that some papers use AP and mAP interchangeably.

## Limitations and Biases#

AP is a great metric that summarizes the PR curve into a single value. Instead of comparing models with a single value of precision and recall at one specific threshold, it lets you compare model performance at every threshold. Although this metric is very popular and commonly used in object detection and information retrieval workflows, it has some limitations. Let's make sure to understand these limitations before using the metric to compare your models.

## AP is often overestimated.

To approximate the area under the curve, it is standard practice to take the maximum precision from the right side of the plot. By doing so, it overestimates the area under the curve.

## AP cannot distinguish between very different-looking PR curves.

Consider the following three plots:

These three plots show very different characteristic, but their APs are exactly the same for all of them. Thus, relying solely on the AP metric is not enough. We recommend plotting the PR curve along with the AP metric to better understand the behavior of your model.

## AP is not confidence score sensitive.

AP uses confidence score to sort inferences, and as long as the sorted order is preserved, the distribution of confidence scores does not change the AP score. Therefore, predictions that have confidence scores within a very small range versus ones with scores that are nicely distributed from 0 to 1 can have the same AP as long as the order is preserved.

## AP uses the interpolated PR curve.

As mentioned in the section above, there are many different ways of interpolating the PR curve. Depending on the granularity of the plot, the AP value can be different, so when comparing models using AP, we need to ensure that it is calculated using the same interpolation method.

## AP is not a fair comparison for thresholded models where the tail part of the PR curve is missing.

It is pretty common for object detectors to filter out predictions with very small confidence scores. In such a scenario, the curve will be missing the tail part, but because the metric considers the entire recall domain, any curves that end early will result in a lower average precision score.

The plot above shows PR curves of two models: one extending to the recall value of `1.0`

and the other one extending
only to `0.6`

. Since a large portion of the area under the curve corresponds to the tail of the curve, model 2
scores a higher AP than model 1.